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Graph Theory

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• Worranat Pakornrat
• Mahindra Jain
• Karleigh Moore
• Satyabrata Dash

Graph theory is the study of mathematical objects known as graphs , which consist of vertices (or nodes ) connected by edges . (In the figure below, the vertices are the numbered circles, and the edges join the vertices.)

3-Cycle " /> A basic graph of 3-Cycle

Any scenario in which one wishes to examine the structure of a network of connected objects is potentially a problem for graph theory. Examples of graph theory frequently arise not only in mathematics but also in physics and computer science.

Terminology

Eulerian graphs, planar graphs, graph coloring.

A non-trivial graph consists of one or more vertices (or nodes ) connected by edges . Each edge connects exactly two vertices, although any given vertex need not be connected by an edge. The degree of a vertex is the number of edges connected to that vertex. In the graph below, vertex \( A \) is of degree 3, while vertices \( B \) and \( C \) are of degree 2. Vertex \( D \) is of degree 1, and vertex \( E \) is of degree 0.

Note: If the degree of each vertex is the same for a graph, we can call that the degree of the graph .

A graph in which it is possible to reach any vertex by traversing the edges from one vertex to another is said to be connected . The set of edges used (not necessarily distinct) is called a path between the given vertices. The graph above is not connected, although there exists a path between any two of the vertices \( A \), \( B \), \( C \), and \( D \).

A graph is said to be complete if there exists an edge connecting every two pairs of vertices. The graph above is not complete but can be made complete by adding extra edges:

Find the number of edges in a complete graph with \( n \) vertices. Finding the number of edges in a complete graph is a relatively straightforward counting problem. Consider the process of constructing a complete graph from \( n \) vertices without edges. One procedure is to proceed one vertex at a time and draw edges between it and all vertices not connected to it. First, \( n-1 \) edges can be drawn between a given vertex and the \( n-1 \) other vertices. Next, \( n-2 \) edges are available between the second vertex and \( n-2 \) other vertices (minus the first, which is already connected). In general, each successive vertex requires one fewer edge to connect than the one right before it. As a result, the total number of edges is \[ (n - 1) + (n - 2) + \cdots + 2 + 1 = \frac{n(n-1)}{2}. \] Equivalently, the number of ways to to select two vertices (for which an edge must exist to connect them) is \[ \dbinom{n}{2} = \frac{n(n-1)}{2}.\ _\square \]

For various applications, it may make sense to give the edges or vertices (or both) some weight . For instance, one can consider a graph consisting of various cities in the United States and edges connecting them representing possible routes between the cities. If one is interested in finding the shortest physical path to travel between the cities, it makes sense to weight the edges by the physical distance between the cities.

Graphs can also be directed or undirected : each edge in a directed graph can point to one or both nodes (for instance, representing one-way travel).

Most of the rest of this article will be concerned with graphs that are connected, unweighted, and undirected.

The project of building 20 roads connecting 9 cities is under way, as outlined above. So far, only some of the 20 roads are constructed, and the digit on each city indicates the number of constructed roads to other cities.

How many complete roads are there among these cities?

There are many types of special graphs. One commonly encountered type is the Eulerian graph , all of whose edges are visited exactly once in a single path. Such a path is known as an Eulerian path . It turns out that it is quite easy to rule out many graphs as non-Eulerian by the following simple rule:

A Eulerian graph has at most two vertices of odd degree.

To see why this fact is true, consider that it is possible to traverse all the edges connected to a vertex of odd degree only if one starts or ends on that vertex during a traversal. Otherwise, one must always enter and exit a given vertex, which uses two edges. However, the entry and exit vertices can be traversed an odd number of times. It is therefore not possible for there to be more than two such vertices, or else one would get "stuck" at some point during an attempted traversal of the graph.

The classic Eulerian graph problem is that of the seven bridges of Königsberg, which Euler solved in 1736.

Seven bridges of Königsberg: The city of Königsberg is connected by seven bridges, as shown. Is it possible to visit all parts of the city by crossing each bridge exactly once? First, we represent the different parts of the city as vertices and each bridge as a vertex connected two parts of the city, as shown below. The graph contains more than two vertices of odd degree, so it is not Eulerian. Therefore, crossing each bridge exactly once is impossible. \(_\square\)

An analogous type of graph is the Hamiltonian path , one in which it is possible to traverse the graph by visiting each vertex exactly once. In general, computing the Hamiltonian path (if one exists) is not a straightforward task.

A graph is said to be planar if it can be drawn on a flat plane without any of the edges crossing. If so, one can define a face of the graph as any region bounded by edges and containing no edges on the interior. One important result regarding planar graphs is as follows:

Euler's Formula Suppose a planar graph has \( V \) vertices, \( F \) faces, and \( E \) edges. Then \[ V - E + F = 2. \]

Let \( K_n \) denote the complete graph with \( n \) vertices. Which of the following is true?

I. \(\hspace{1mm} K_4 \) is planar. II. \(\hspace{1mm} K_5 \) is planar. III. \(\hspace{1mm} K_6 \) is planar.

One important problem in graph theory is that of graph coloring . Suppose each vertex in a graph is assigned a color such that no two adjacent vertices share the same color. Clearly, it is possible to color every graph in this way: in the worst case, one could simply use a number of colors equal to the number of vertices. (Indeed, for a complete graph, the minimum number of colors is equal to the number of vertices.) Of particular interest is the minimum number of colors \( k \) needed to avoid connecting vertices of like color, which is known as the chromatic number \( k \) of the graph. Equivalently, the graph is said to be \( k \)-colorable.

In particular, when coloring a map, generally one wishes to avoid coloring the same color two countries that share a border. The problem of map coloring neatly reduces to a graph coloring problem: simply represent each country by a vertex, with an edge connecting each pair of countries that share a border. In 1976, Appel and Haken proved the four color theorem , which holds that no graph corresponding to a map has a chromatic number greater than 4.

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4.E: Graph Theory (Exercises)

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4.1: Definitions

If 10 people each shake hands with each other, how many handshakes took place? What does this question have to do with graph theory?

This is asking for the number of edges in \(K_{10}\text{.}\) Each vertex (person) has degree (shook hands with) 9 (people). So the sum of the degrees is \(90\text{.}\) However, the degrees count each edge (handshake) twice, so there are 45 edges in the graph. That is how many handshakes took place.

Among a group of 5 people, is it possible for everyone to be friends with exactly 2 of the people in the group? What about 3 of the people in the group?

It is possible for everyone to be friends with exactly 2 people. You could arrange the 5 people in a circle and say that everyone is friends with the two people on either side of them (so you get the graph \(C_5\)). However, it is not possible for everyone to be friends with 3 people. That would lead to a graph with an odd number of odd degree vertices which is impossible since the sum of the degrees must be even.

Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not.

Yes. For example, both graphs below contain 6 vertices, 7 edges, and have degrees (2,2,2,2,3,3).

Are the two graphs below equal? Are they isomorphic? If they are isomorphic, give the isomorphism. If not, explain.

• Graph 1: \(V = \{a,b,c,d,e\}\text{,}\) \(E = \{\{a,b\}, \{a,c\}, \{a,e\}, \{b,d\}, \{b,e\}, \{c,d\}\}\text{.}\)

The graphs are not equal. For example, graph 1 has an edge \(\{a,b\}\) but graph 2 does not have that edge. They are isomorphic. One possible isomorphism is \(f:G_1 \to G_2\) defined by \(f(a) = d\text{,}\) \(f(b) = c\text{,}\) \(f(c) = e\text{,}\) \(f(d) = b\text{,}\) \(f(e) = a\text{.}\)

Consider the following two graphs:

• \(V_1=\{a,b,c,d,e,f,g\}\)
• \(E_1=\{\{a,b\},\{a,d\},\{b,c\},\{b,d\},\{b,e\},\{b,f\},\{c,g\},\{d,e\},\)
• \(\{e,f\},\{f,g\}\}\text{.}\)
• \(V_2=\{v_1,v_2,v_3,v_4,v_5,v_6,v_7\}\text{,}\)
• \(E_2=\{\{v_1,v_4\},\{v_1,v_5\},\{v_1,v_7\},\{v_2,v_3\},\{v_2,v_6\},\)
• \(\{v_3,v_5\},\{v_3,v_7\},\{v_4,v_5\},\{v_5,v_6\},\{v_5,v_7\}\}\)

Does \(f\) define an isomorphism between Graph 1 and Graph 2? Explain.

• Define a new function \(g\) (with \(g\not=f\)) that defines an isomorphism between Graph 1 and Graph 2.
• Is the graph pictured below isomorphic to Graph 1 and Graph 2? Explain.

Which of the graphs below are bipartite? Justify your answers.

Three of the graphs are bipartite. The one which is not is \(C_7\) (second from the right). To see that the three graphs are bipartite, we can just give the bipartition into two sets \(A\) and \(B\text{,}\) as labeled below:

The graph \(C_7\) is not bipartite because it is an odd cycle. You would want to put every other vertex into the set \(A\text{,}\) but if you travel clockwise in this fashion, the last vertex will also be put into the set \(A\text{,}\) leaving two \(A\) vertices adjacent (which makes it not a bipartition).

For which \(n \ge 3\) is the graph \(C_n\) bipartite?

For each of the following, try to give two different unlabeled graphs with the given properties, or explain why doing so is impossible.

• Two different trees with the same number of vertices and the same number of edges. A tree is a connected graph with no cycles.
• Two different graphs with 8 vertices all of degree 2.
• Two different graphs with 5 vertices all of degree 4.
• Two different graphs with 5 vertices all of degree 3.
• For example:
• This is not possible if we require the graphs to be connected. If not, we could take \(C_8\) as one graph and two copies of \(C_4\) as the other.
• Not possible. If you have a graph with 5 vertices all of degree 4, then every vertex must be adjacent to every other vertex. This is the graph \(K_5\text{.}\)
• This is not possible. In fact, there is not even one graph with this property (such a graph would have \(5\cdot 3/2 = 7.5\) edges).

4.2: Planar Graphs

Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain.

No. A (connected) planar graph must satisfy Euler's formula: \(v - e + f = 2\text{.}\) Here \(v - e + f = 6 - 10 + 5 = 1\text{.}\)

The graph \(G\) has 6 vertices with degrees \(2, 2, 3, 4, 4, 5\text{.}\) How many edges does \(G\) have? Could \(G\) be planar? If so, how many faces would it have. If not, explain.

\(G\) has 10 edges, since \(10 = \frac{2+2+3+4+4+5}{2}\text{.}\) It could be planar, and then it would have 6 faces, using Euler's formula: \(6-10+f = 2\) means \(f = 6\text{.}\) To make sure that it is actually planar though, we would need to draw a graph with those vertex degrees without edges crossing. This can be done by trial and error (and is possible).

I'm thinking of a polyhedron containing 12 faces. Seven are triangles and four are quadralaterals. The polyhedron has 11 vertices including those around the mystery face. How many sides does the last face have?

Say the last polyhedron has \(n\) edges, and also \(n\) vertices. The total number of edges the polyhedron has then is \((7 \cdot 3 + 4 \cdot 4 + n)/2 = (37 + n)/2\text{.}\) In particular, we know the last face must have an odd number of edges. We also have that \(v = 11 \text{.}\) By Euler's formula, we have \(11 - (37+n)/2 + 12 = 2\text{,}\) and solving for \(n\) we get \(n = 5\text{,}\) so the last face is a pentagon.

Consider some classic polyhedrons.

• An octahedron is a regular polyhedron made up of 8 equilateral triangles (it sort of looks like two pyramids with their bases glued together). Draw a planar graph representation of an octahedron. How many vertices, edges and faces does an octahedron (and your graph) have?
• The traditional design of a soccer ball is in fact a (spherical projection of a) truncated icosahedron. This consists of 12 regular pentagons and 20 regular hexagons. No two pentagons are adjacent (so the edges of each pentagon are shared only by hexagons). How many vertices, edges, and faces does a truncated icosahedron have? Explain how you arrived at your answers. Bonus: draw the planar graph representation of the truncated icosahedron.
• Your “friend” claims that he has constructed a convex polyhedron out of 2 triangles, 2 squares, 6 pentagons and 5 octagons. Prove that your friend is lying. Hint: each vertex of a convex polyhedron must border at least three faces.

Prove Euler's formula using induction on the number of edges in the graph.

Let \(P(n)\) be the statement, “every planar graph containing \(n\) edges satisfies \(v - n + f = 2\text{.}\)” We will show \(P(n)\) is true for all \(n \ge 0\text{.}\) Base case: there is only one graph with zero edges, namely a single isolated vertex. In this case \(v = 1\text{,}\) \(f = 1\) and \(e = 0\text{,}\) so Euler's formula holds. Inductive case: Suppose \(P(k)\) is true for some arbitrary \(k \ge 0\text{.}\) Now consider an arbitrary graph containing \(k+1\) edges (and \(v\) vertices and \(f\) faces). No matter what this graph looks like, we can remove a single edge to get a graph with \(k\) edges which we can apply the inductive hypothesis to. There are two possibilities. First, the edge we remove might be incident to a degree 1 vertex. In this case, also remove that vertex. The smaller graph will now satisfy \(v-1 - k + f = 2\) by the induction hypothesis (removing the edge and vertex did not reduce the number of faces). Adding the edge and vertex back gives \(v - (k+1) + f = 2\text{,}\) as required. The second case is that the edge we remove is incident to vertices of degree greater than one. In this case, removing the edge will keep the number of vertices the same but reduce the number of faces by one. So by the inductive hypothesis we will have \(v - k + f-1 = 2\text{.}\) Adding the edge back will give \(v - (k+1) + f = 2\) as needed. Therefore, by the principle of mathematical induction, Euler's formula holds for all planar graphs.

Prove Euler's formula using induction on the number of vertices in the graph.

Euler's formula (\(v - e + f = 2\)) holds for all connected planar graphs. What if a graph is not connected? Suppose a planar graph has two components. What is the value of \(v - e + f\) now? What if it has \(k\) components?

Prove that the Petersen graph (below) is not planar.

What is the length of the shortest cycle? (This quantity is usually called the girth of the graph.)

Prove that any planar graph with \(v\) vertices and \(e\) edges satisfies \(e \le 3v - 6\text{.}\)

We know in any planar graph the number of faces \(f\) satisfies \(3f \le 2e\) since each face is bounded by at least three edges, but each edge borders two faces. Combine this with Euler's formula:

\begin{equation*} v - e + f = 2 \end{equation*} \begin{equation*} v - e + \frac{2e}{3} \ge 2 \end{equation*} \begin{equation*} 3v - e \ge 6 \end{equation*} \begin{equation*} 3v - 6 \ge e. \end{equation*}

Prove that any planar graph must have a vertex of degree 5 or less.

4.3: Coloring

What is the smallest number of colors you need to properly color the vertices of \(K_{4,5}\text{?}\) That is, find the chromatic number of the graph.

2, since the graph is bipartite. One color for the top set of vertices, another color for the bottom set of vertices.

Draw a graph with chromatic number 6 (i.e., which requires 6 colors to properly color the vertices). Could your graph be planar? Explain.

For example, \(K_6\text{.}\) If the chromatic number is 6, then the graph is not planar; the 4-color theorem states that all planar graphs can be colored with 4 or fewer colors.

Find the chromatic number of each of the following graphs.

The chromatic numbers are 2, 3, 4, 5, and 3 respectively from left to right.

A group of 10 friends decides to head up to a cabin in the woods (where nothing could possibly go wrong). Unfortunately, a number of these friends have dated each other in the past, and things are still a little awkward. To get the cabin, they need to divide up into some number of cars, and no two people who dated should be in the same car.

• What is the smallest number of cars you need if all the relationships were strictly heterosexual? Represent an example of such a situation with a graph. What kind of graph do you get?
• What do these questions have to do with coloring?

What is the smallest number of colors that can be used to color the vertices of a cube so that no two adjacent vertices are colored identically?

The cube can be represented as a planar graph and colored with two colors as follows:

Since it would be impossible to color the vertices with a single color, we see that the cube has chromatic number 2 (it is bipartite).

Prove the chromatic number of any tree is two. Recall, a tree is a connected graph with no cycles.

• Describe a procedure to color the tree below.
• The chromatic number of \(C_n\) is two when \(n\) is even. What goes wrong when \(n\) is odd?
• Prove that your procedure from part (a) always works for any tree.
• Now, prove using induction that every tree has chromatic number 2.

Prove the 6-color theorem: every planar graph has chromatic number 6 or less. Do not assume the 4-color theorem (whose proof is MUCH harder), but you may assume the fact that every planar graph contains a vertex of degree at most 5.

Not all graphs are perfect. Give an example of a graph with chromatic number 4 that does not contain a copy of \(K_4\text{.}\) That is, there should be no 4 vertices all pairwise adjacent.

The wheel graph below has this property. The outside of the wheel forms an odd cycle, so requires 3 colors, the center of the wheel must be different than all the outside vertices.

Prove by induction on vertices that any graph \(G\) which contains at least one vertex of degree less than \(\Delta(G)\) (the maximal degree of all vertices in \(G\)) has chromatic number at most \(\Delta(G)\text{.}\)

You have a set of magnetic alphabet letters (one of each of the 26 letters in the alphabet) that you need to put into boxes. For obvious reasons, you don't want to put two consecutive letters in the same box. What is the fewest number of boxes you need (assuming the boxes are able to hold as many letters as they need to)?

If we drew a graph with each letter representing a vertex, and each edge connecting two letters that were consecutive in the alphabet, we would have a graph containing two vertices of degree 1 (A and Z) and the remaining 24 vertices all of degree 2 (for example, \(D\) would be adjacent to both \(C\) and \(E\)). By Brooks' theorem, this graph has chromatic number at most 2, as that is the maximal degree in the graph and the graph is not a complete graph or odd cycle. Thus only two boxes are needed.

Prove that if you color every edge of \(K_6\) either red or blue, you are guaranteed a monochromatic triangle (that is, an all red or an all blue triangle).

4.4: Euler Paths and Circuits

You and your friends want to tour the southwest by car. You will visit the nine states below, with the following rather odd rule: you must cross each border between neighboring states exactly once (so, for example, you must cross the Colorado-Utah border exactly once). Can you do it? If so, does it matter where you start your road trip? What fact about graph theory solves this problem?

This is a question about finding Euler paths. Draw a graph with a vertex in each state, and connect vertices if their states share a border. Exactly two vertices will have odd degree: the vertices for Nevada and Utah. Thus you must start your road trip at in one of those states and end it in the other.

Which of the following graphs contain an Euler path? Which contain an Euler circuit?

• \(K_5\text{.}\)
• \(K_{5,7}\)
• \(K_{2,7}\)
• \(K_4\) does not have an Euler path or circuit.
• \(K_5\) has an Euler circuit (so also an Euler path).
• \(K_{5,7}\) does not have an Euler path or circuit.
• \(K_{2,7}\) has an Euler path but not an Euler circuit.
• \(C_7\) has an Euler circuit (it is a circuit graph!)
• \(P_7\) has an Euler path but no Euler circuit.

Edward A. Mouse has just finished his brand new house. The floor plan is shown below:

• Edward wants to give a tour of his new pad to a lady-mouse-friend. Is it possible for them to walk through every doorway exactly once? If so, in which rooms must they begin and end the tour? Explain.
• Is it possible to tour the house visiting each room exactly once (not necessarily using every doorway)? Explain.
• After a few mouse-years, Edward decides to remodel. He would like to add some new doors between the rooms he has. Of course, he cannot add any doors to the exterior of the house. Is it possible for each room to have an odd number of doors? Explain.

For which \(n\) does the graph \(K_n\) contain an Euler circuit? Explain.

When \(n\) is odd, \(K_n\) contains an Euler circuit. This is because every vertex has degree \(n-1\text{,}\) so an odd \(n\) results in all degrees being even.

For which \(m\) and \(n\) does the graph \(K_{m,n}\) contain an Euler path? An Euler circuit? Explain.

If both \(m\) and \(n\) are even, then \(K_{m,n}\) has an Euler circuit. When both are odd, there is no Euler path or circuit. If one is 2 and the other is odd, then there is an Euler path but not an Euler circuit.

For which \(n\) does \(K_n\) contain a Hamilton path? A Hamilton cycle? Explain.

Add texts here. Do not delete this text first.

All values of \(n\text{.}\) In particular, \(K_n\) contains \(C_n\) as a subgroup, which is a cycle that includes every vertex.

For which \(m\) and \(n\) does the graph \(K_{m,n}\) contain a Hamilton path? A Hamilton cycle? Explain.

As long as \(|m-n| \le 1\text{,}\) the graph \(K_{m,n}\) will have a Hamilton path. To have a Hamilton cycle, we must have \(m=n\text{.}\)

A bridge builder has come to Königsberg and would like to add bridges so that it is possible to travel over every bridge exactly once. How many bridges must be built?

If we build one bridge, we can have an Euler path. Two bridges must be built for an Euler circuit.

We are looking for a Hamiltonian cycle, and this graph does have one:

• Suppose a graph has a Hamilton path. What is the maximum number of vertices of degree one the graph can have? Explain why your answer is correct.
• Find a graph which does not have a Hamilton path even though no vertex has degree one. Explain why your example works.

Consider the following graph:

• Find a Hamilton path. Can your path be extended to a Hamilton cycle?
• Is the graph bipartite? If so, how many vertices are in each “part”?
• Use your answer to part (b) to prove that the graph has no Hamilton cycle.
• Suppose you have a bipartite graph \(G\) in which one part has at least two more vertices than the other. Prove that \(G\) does not have a Hamilton path.

4.5: Matching in Bipartite Graphs

Find a matching of the bipartite graphs below or explain why no matching exists.

The first and third graphs have a matching, shown in bold (there are other matchings as well). The middle graph does not have a matching. If you look at the three circled vertices, you see that they only have two neighbors, which violates the matching condition \(\card{N(S)} \ge \card{S}\) (the three circled vertices form the set \(S\)).

A bipartite graph that doesn't have a matching might still have a partial matching . By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). Every bipartite graph (with at least one edge) has a partial matching, so we can look for the largest partial matching in a graph.

Your “friend” claims that she has found the largest partial matching for the graph below (her matching is in bold). She explains that no other edge can be added, because all the edges not used in her partial matching are connected to matched vertices. Is she correct?

One way you might check to see whether a partial matching is maximal is to construct an alternating path . This is a sequence of adjacent edges, which alternate between edges in the matching and edges not in the matching (no edge can be used more than once). If an alternating path starts and stops with an edge not in the matching, then it is called an augmenting path .

• Find the largest possible alternating path for the partial matching of your friend's graph. Is it an augmenting path? How would this help you find a larger matching?
• Find the largest possible alternating path for the partial matching below. Are there any augmenting paths? Is the partial matching the largest one that exists in the graph?

The two richest families in Westeros have decided to enter into an alliance by marriage. The first family has 10 sons, the second has 10 girls. The ages of the kids in the two families match up. To avoid impropriety, the families insist that each child must marry someone either their own age, or someone one position younger or older. In fact, the graph representing agreeable marriages looks like this:

The question: how many different acceptable marriage arrangements which marry off all 20 children are possible?

• How many marriage arrangements are possible if we insist that there are exactly 6 boys marry girls not their own age?
• Could you generalize the previous answer to arrive at the total number of marriage arrangements?
• How do you know you are correct? Try counting in a different way. Look at smaller family sizes and get a sequence.
• Can you give a recurrence relation that fits the problem?

We say that a set of vertices \(A \subseteq V\) is a vertex cover if every edge of the graph is incident to a vertex in the cover (so a vertex cover covers the edges ). Since \(V\) itself is a vertex cover, every graph has a vertex cover. The interesting question is about finding a minimal vertex cover, one that uses the fewest possible number of vertices.

• Suppose you had a matching of a graph. How can you use that to get a minimal vertex cover? Will your method always work?
• Suppose you had a minimal vertex cover for a graph. How can you use that to get a partial matching? Will your method always work?
• What is the relationship between the size of the minimal vertex cover and the size of the maximal partial matching in a graph?

For many applications of matchings, it makes sense to use bipartite graphs. You might wonder, however, whether there is a way to find matchings in graphs in general.

• For which \(n\) does the complete graph \(K_n\) have a matching?
• Prove that if a graph has a matching, then \(\card{V}\) is even.
• Is the converse true? That is, do all graphs with \(\card{V}\) even have a matching?
• What if we also require the matching condition? Prove or disprove: If a graph with an even number of vertices satisfies \(\card{N(S)} \ge \card{S}\) for all \(S \subseteq V\text{,}\) then the graph has a matching.

Browse Course Material

Course info.

• Prof. Yufei Zhao

Departments

• Mathematics

As Taught In

• Applied Mathematics
• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Graph theory and additive combinatorics, assignments.

The problem set (PDF) is a list of problems for practice. A subset of these problems will be designated as to-be-turned in. Only the designated problems are required to be submitted. Bonus problems, marked by ★, are more challenging. You may attain a grade of A- by only solving the non-starred problems. To attain a grade of A or A+, you should solve a substantial number of starred problems.

Study Guides > Mathematics for the Liberal Arts

Assignment: graph theory.

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Graph Theory: Word Document

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• Graph Theory Writing Assignment. Authored by: Lippman, David. License: CC BY: Attribution .

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• Graph theory

A branch of discrete mathematics, distinguished by its geometric approach to the study of various objects. The principal object of the theory is a graph and its generalizations. The first problems in the theory of graphs were solutions of mathematical puzzles (the problem of the bridges of Königsberg, the disposition of queens on a chessboard, transportation problems, the travelling-salesman problem, etc.). One of the first results in graph theory was a criterion on the possibility of traversing all edges of a graph without passing through any edge more than once; it was obtained by L. Euler in 1736 in solving the problem of the bridges of Königsberg. The four-colour problem , formulated in the mid-19th century, though a mere amusement puzzle at first sight, led to studies of graphs of both theoretical and applied interest. Certain studies in the mid-19th century contain results of importance to graph theory, obtained by solving practical problems. Thus, G. Kirchhoff's complete set of equations [2] for currents and voltages in electric circuits really amounts to representing the circuit by a graph with skeleton trees, with the aid of which linearly independent circuit systems are obtained. A. Cayley [3] , starting from the problem of calculating the number of isomers of saturated hydrocarbons, arrived at the problems of listing and describing trees with certain properties, and solved some of them. In the 20th century, problems involving graphs began to arise not only in physics, chemistry, electrical engineering, biology, economics, sociology, etc., but also in mathematics itself — in topology, algebra, probability theory, and number theory. At the beginning of the 20th century, graphs were used to represent certain mathematical objects and to formally state various discrete problems; besides the term "graph" , other terms such as map, complex, diagram, network, labyrinth, were also employed. Following the appearance of the monograph of D. König [4] in 1936, the term "graph" has been used more frequently than other terms. This study reviewed the facts known at that time. The first results, concerning connectivity properties, planarity, and graph symmetry, which paved the way for a number of novel directions of study in graph theory, appeared in the 1920s and 1930s. The scope of research in graph theory was considerably extended in the late 1940s and early 1950s, mainly as a result of the development of cybernetics and calculation techniques. Interest in graph theory increased, and the range of problems dealt with by the theory was considerably extended. The use of electronic computers made it possible to solve practical problems involving extensive calculations, which could not be solved previously. Methods were developed for solving a number of extremal problems in graph theory; one such problem is the construction of the maximum flow across a network (cf. Flow in a network ). It was shown, for individual classes of graphs (trees, planar graphs, etc.), which had been studied before, that the solution of certain problems was simpler for such graphs than for arbitrary graphs (finding conditions for the existence of graphs with certain properties, establishment of graph isomorphism, etc.).

Some problems in graph theory are more combinatorial in nature, while others are essentially geometric. The former include, for example, the calculation and listing of graphs with prescribed properties and the construction of graphs with prescribed properties. Many types of problems, such as on graph circuits (cf. Graph circuit ) and graph imbedding , are of geometric (topological) nature. Certain problems concern the modes of classification of graphs, e.g. classification by the properties of partitions of them. Results concerning the existence of graphs with certain properties may be exemplified by the criterion of realizability of numbers by the degrees of the vertices of a given graph: A collection of integers $ 0 \leq d _ {1} \leq \dots \leq d _ {n} $, the sum of which is even, can be realized by degrees of the vertices of a graph without loops and multiple edges if and only if the condition

$$ \sum _ {i = 1 } ^ { r } d _ {i} \leq r ( r - 1) + \sum _ {i = r + 1 } ^ { n } \min ( r, d _ {i} ) $$

is fulfilled for any $ r $( $ 1 \leq r \leq n - 1 $).

Problems of the enumeration of graphs with prescribed properties can be exemplified by problems of finding the number of non-isomorphic graphs with the same number of vertices and/or edges. The number of non-isomorphic trees with $ n $ vertices is given by the asymptotic formula

$$ t _ {n} = \ C \frac{\theta ^ {n} }{n ^ {5/2} } + O \left ( \frac{\theta ^ {n} }{n ^ {7/2} } \right ) , $$

where $ C = 0.534948 \dots $, $ \theta = 2.95576 \dots $. The number $ g _ {n} $ of non-isomorphic graphs without loops or multiple edges and with $ n $ vertices is given by the formula

$$ g _ {n} = \ \frac{2 ^ {( _ {2} ^ {n} ) } }{n! } \left ( 1 + \frac{2n ( n - 1) }{2 ^ {n} } + \frac{8 ( 3n - 7) n! }{3 ( n - 3) ! \cdot 2 ^ {2n} } + O \left ( \frac{n ^ {5} }{2 ^ {5n/2} } \right ) \right ) . $$

Graph theory deals with specific types of problems, as well as with problems of a general nature. One type of such specific problems is the connectivity of graphs, and the study of the structure of a graph based on its connectivity (cf. Graph, connectivity of a ). In the analysis of the reliability of electronic circuits or communications networks there arises the problem of finding the number of non-intersecting chains connecting the various vertices of a graph. Here, a number of results have been obtained. Thus, the smallest number of vertices separating two non-adjacent vertices of a graph is equal to the greatest number of non-intersecting (at the vertices) simple chains which connect this pair of vertices. Criteria have been found and effective algorithms have been developed for the establishment of the degree of connectivity of a graph (the smallest number of vertices or edges the removal of which destroys the connectedness of the graph).

Another direction of study in graph theory is the investigation of walks (edge progressions) containing all the vertices or all the edges of a graph (cf. Graph circuit ). A simple criterion for the existence of a walk containing all the edges of a graph is available: In a connected graph a cycle containing all the edges and passing through each edge once and only once exists if and only if the degrees of all except two vertices of the graph are even. If the set of vertices of a graph is traversed, only a number of sufficient conditions for the existence of a cycle passing through each vertex once is available.

A characteristic type of problems dealt with by graph theory concerns graph colouring. They involve the partition of the set of vertices (edges) having certain properties, e.g. adjacent vertices (edges) are to belong to different sets (vertices or edges of the same set are drawn with the same colour; cf. Graph colouring ). It has been proved that the least number of colours required for colouring the edges of any graph without loops of maximum degree $ \sigma $ is $ [ 3 \sigma /2] $, while $ \sigma + 1 $ colours are sufficient for the colouring of the vertices of any graph without loops or multiple edges.

Other types of problems also exist (cf. Covering and packing ; Graph imbedding ; Graph, numerical characteristics of a ); some of them were suggested by other branches of mathematics. Thus, topology initiated the study of imbedding graphs in various surfaces. For example, the following condition is necessary and sufficient for the imbedding of a graph in a plane (the Pontryagin–Kuratowski criterion): A graph is planar (cf. Graph, planar ) if and only if it contains no subgraphs obtained by partitioning of the edges of the complete $ 5 $- vertex graph, or the complete bipartite graph, with three vertices in each part. Algebra initiated the study of graph automorphisms (cf. Graph automorphism ). It has been shown, in particular, that any finite group is isomorphic to the automorphism group of some graph. Probability theory stimulated the study of random graphs (cf. Graph, random ). Many properties have been studied for "almost-all" graphs; it has been shown, for example, that almost-all graphs with $ n $ vertices are connected, that their diameter is 2, and that they possess a Hamilton cycle (a cycle passing through each vertex of the graph once).

Graph theory comprises specific methods for solving extremal problems. One such method is based on the max-flow-min-cut theorem , which states that the maximum flow which can pass through a network from a vertex $ u $ to a vertex $ v $ equals the minimum transmission capacity of the cuts separating the vertices $ u $ and $ v $( cf. Flow in a network ). Various effective algorithms for finding the maximum flow have been developed.

Algorithmic problems are of major importance in graph theory. For finite graphs, i.e. for graphs with a finite number of vertices and edges, an algorithm for a solution of problems, including extremal problems, usually exists. Many problems involving finite graphs can be solved by complete inspection of all admissible variants. However, such a method solves the problem only for graphs with a small number of vertices and edges. It is therefore highly important in graph theory to construct effective algorithms by means of which an exact or an approximate solution can be found. Such algorithms have in fact been constructed for certain problems, such as the establishment of planarity of graphs, the determination of isomorphism of trees, or the finding of the maximum flow.

The results and methods of graph theory are used for solving transportation problems, to find optimal solutions for the assignment problem, to identify bottlenecks in the planning and control of project development, in establishing the optimum routes for the supply of goods, as well as in modelling complex technological processes, in the construction of various discrete mechanisms, in programming, etc.

Regarding terminology in graph theory see also Graph . For a good historical account of graph theory see, e.g., [a1] . The Pontryagin–Kuratowski criterion is usually known as Kuratowski's theorem.

• This page was last edited on 15 March 2023, at 20:08.
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Graph Theory Tutorial

Basics of graph theory.

• Graph and its representations
• Mathematics | Graph Theory Basics - Set 1
• Types of Graphs with Examples
• Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph
• Graph measurements: length, distance, diameter, eccentricity, radius, center
• Articulation Points (or Cut Vertices) in a Graph
• Bridges in a graph
• Mathematics | Independent Sets, Covering and Matching
• How to find Shortest Paths from Source to all Vertices using Dijkstra's Algorithm
• Depth First Search or DFS for a Graph
• Breadth First Search or BFS for a Graph
• Introduction to Tree - Data Structure and Algorithm Tutorials
• Prim’s Algorithm for Minimum Spanning Tree (MST)
• Kruskal’s Minimum Spanning Tree (MST) Algorithm
• Huffman Coding | Greedy Algo-3
• Tree Traversal Techniques - Data Structure and Algorithm Tutorials
• Travelling Salesman Problem using Dynamic Programming

Special Graph

• Check whether a given graph is Bipartite or not
• Eulerian Graphs
• Eulerian path and circuit for undirected graph
• Fleury's Algorithm for printing Eulerian Path or Circuit
• Chinese Postman or Route Inspection | Set 1 (introduction)
• Mathematics | Matching (graph theory)
• Approximation Algorithms
• Introduction to Graph Coloring
• Edge Coloring of a Graph

Planar Graph

• Mathematics | Planar Graphs and Graph Coloring

Directed Graphs

• Degree Centrality (Centrality Measure)
• Check if a graph is Strongly, Unilaterally or Weakly connected
• Strongly Connected Components
• Mathematics | Euler and Hamiltonian Paths
• Tarjan's Algorithm to find Strongly Connected Components
• Handshaking Lemma and Interesting Tree Properties

Group Theory

• Algebraic Structure
• Homomorphism & Isomorphism of Group
• Group Isomorphisms and Automorphisms
• Mathematics | Rings, Integral domains and Fields

Graph Theory is a branch of mathematics that is concerned with the study of relationships between different objects. A graph is a collection of various vertexes also known as nodes, and these nodes are connected with each other via edges. In this tutorial, we have covered all the topics of Graph Theory like characteristics, eulerian graphs, planar graphs, special graphs, trees, paths in graph theory, etc. This Graph theory tutorial will be helpful in learning the concept of the subject along with the applications of graph theory in real life and in various fields. 

• Introduction to Graph
• Basic Terminology of a Graph
• Types of a Graph
• Walks, Trails, Paths, and Circuits
• Graph Distance Components
• Cut-Vertices and Cut-Edges
• Bridge in Graph
• Independent Sets
• Shortest Path Algorithms [Dijsktra’s Algorithm]
• Application of Graph Theory
• Graph Traversals[DFS]
• Graph Traversals[BFS]
• Characterizations of Trees
• Prim’s Minimum Spanning Tree 
• Kruskal’s Minimum Spanning Tree
• Huffman Codes 
• Tree Traversals
• Traveling Salesman Problem
• Bipartite Graphs  
• Independent Sets and Covering
• Eulerian Graphs- Fleury’s Algorithm
• Eulerian Graphs- Chinese-Postman-Problem Hamilton
• Matching- Basics, Perfect, Bipartite
• Chromatic Numbers, Greedy Coloring Algorithm
• Edge Coloring
• Planar Graph- Basics, Planarity Testing
• Directed Graphs- Degree Centrality
• Directed Graphs- Weak Connectivity
• Directed Graphs- Strong Components 
• Directed Graphs- Eulerian, Hamilton Directed Graphs
• Directed Graphs- Tarjans’ Algorithm To Find Strongly Connected Component
• Handshaking in Graph Theorem
• Groups, Subgroups, Semi Groups
• Isomorphism, Homomorphism
• Automorphism
• Rings, Integral Domains, Fields

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Graph Theory

In mathematics and computer science, Graph theory is the study of points and lines. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines or arcs, called edges. Graphs in this context differ from the more familiar coordinate plots that portray mathematical relations and functions. Graph theory has proven useful in the design of integrated circuits for computers and other electronic devices.

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Module 4: Graph Theory

Assignment: graph theory.

A spell checker in a word processing program makes suggestions when it finds a word not in the dictionary. To determine what words to suggest, it tries to find similar words. One measure of word similarity is the Levenshtein distance, which measures the number of substitutions, additions, or deletions that are required to change one word into another.

For example, the words spit and spot are a distance of 1 apart; changing spit to spot requires one substitution (i for o). Likewise, spit is distance 1 from pit since the change requires one deletion (the s). The word spite is also distance 1 from spit since it requires one addition (the e). The word soot is distance 2 from spit since two substitutions would be required.

a) Create a graph using words as vertices, and edges connecting words with a Levenshtein distance of 1. Use the misspelled word “moke” as the center, and try to find at least 10 connected dictionary words.

b) How might a spell checker use this graph?

Here’s some ideas on how to upload a graph:

• Draw it on paper, scan it into the computer, and upload it. If your file is too large to upload, you are welcome to email it to me instead.
• Draw it on paper, take a picture with your digital camera or cell phone camera and upload it. Please make sure it’s legible in the picture!
• Draw it in Paint or another computer drawing program and save it as a picture and upload it.
• In Word, use the drawing tools to create a graph, perhaps using Text Boxes and Lines. Save and upload

If you really can’t figure out a good way to digitally send your graph, you can physically drop it by my office.

Here is an example of this type of graph, using “rab” as the center. Pardon the quality; I drew it in Paint :)

Download the assignment from one of the links below (.docx or .rtf):

Graph Theory: Word Document

Graph Theory: Rich Text Format

• Mathematics
• NOC:Graph Theory (Video) 
• Co-ordinated by : IISER Pune
• Available from : 2017-06-08
• Intro Video
• Basic Concepts
• Basic Concepts 1
• Eulerian and Hamiltonian Graph
• Eulerian and Hamiltonian Graph 1
• Bipartite Graph
• Diameter of a graph; Isomorphic graphs
• Minimum Spanning Tree
• Minimum Spanning Trees (Cont.)
• Minimum Spanning Trees (Cont..)
• Minimum Spanning Trees (Cont...)
• Maximum Matching in Bipartite Graph
• Maximum Matching in Bipartite Graph 1
• Hall's Theorem and Konig's Theorem
• Hall's Theorem and Konig's Theorem 1
• Independent Set and Edge Cover
• Independent Set and Edge Cover 1
• Matching in General Graphs
• Proof of Halls Theorem
• Stable Matching
• Gale-Shapley Algorithm
• Graph Connectivity
• Graph Connectivity 1
• 2-Connected Graphs
• 2-Connected Graphs 1
• Subdivision of an edge; 2-edge-connected graphs
• Problems Related to Graphs Connectivity
• Flow Network
• Residual Network and Augmenting Path
• Augmenting Path Algorithm
• Max-Flow and Min-Cut
• Max-Flow and Min-Cut Theorem
• Vertex Colouring
• Chromatic Number and Max. Degree
• Edge Colouring
• Planar Graphs & Euler's Formula
• Characterization Of Planar Graphs
• Colouring of Planar Graphs
• Watch on YouTube
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• Study Guides
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DPABINet: A turn-key brain network and graph theory analysis platform based on MRI data

by Science China Press

DPABINet, developed by Dr. Chao-Gan Yan's team at the Institute of Psychology, Chinese Academy of Sciences, simplifies brain network analysis with a user-friendly, one-click software that requires no programming skills.

Leveraging the widely-used DPARSF/DPABI/DPABISurf platform, it democratizes access to complex brain imaging analyses. This innovation is set to advance brain function research and clinical studies by making sophisticated analyses more accessible to a broader range of researchers. The work is published in the journal Science Bulletin .

The human brain , a marvel of complexity, operates as an intricate network where various regions collaborate structurally and functionally. These interactions forge complex patterns that underpin the diverse functionalities of the brain.

Understanding these complex functions necessitates a deep dive into the brain's networks and the sophisticated interconnections and communication modalities they encompass. Moreover, the exploration of brain network mechanisms opens new vistas in the study of diseases marked by abnormalities in brain function, such as brain injuries and mental disorders , making the investigation of the brain's complex network system paramount for a holistic grasp of its functionalities.

Graph Theory, the bedrock of network science, has found extensive application in dissecting the attributes of complex networks, including those of the brain. It lays out a framework for examining the topological structures of these networks, unveiling pivotal organizational details of functional brain networks across both micro and macro scales.

Despite graph theory's immense analytical potential, the intricacy of existing professional imaging data processing software—often demanding high-level programming acumen—has been a bottleneck, hampering the broader adoption and application of these analytical methods.

Addressing this challenge, Dr. Chao-Gan Yan and his team from the Magnetic Resonance Imaging Research Center at the Institute of Psychology, Chinese Academy of Sciences, have innovated beyond their previously acclaimed brain imaging processing platform (DPARSF/DPABI/DPABISurf, cited in over 5,000 research works, with the DPABI software paper being distinguished as one of the top 0.01% highly cited papers by ESI and featured among the hot papers in the Chinese medical field for 2015-2019).

They unveiled DPABINet, a pioneering platform that amalgamates cutting-edge image processing modules like Brain Connectivity Toolbox, FSLNets, BrainNet Viewer, Circos, SPM, PALM, among others, leveraging docker technology to furnish a user-friendly, cross-platform interface and algorithms.

DPABINet's intuitive graphical interface (GUI) empowers users to seamlessly construct brain networks, conduct graph theory analysis, and perform statistical analysis and result visualization with a single click, eliminating the need for programming or scripting expertise.

Additionally, DPABINet enhances the research on brain structural networks derived from diffusion-weighted imaging, providing a robust analysis framework for brain structural fiber networks predicated on DPABIFiber's preprocessing results.

Inheriting and broadening the design ethos of DPARSF/DPABI/DPABISurf, DPABINet significantly lowers the barrier to entry, enabling users devoid of programming knowledge to undertake precise and efficient brain network construction, graph theory analysis, and data interpretation.

To bolster user proficiency, Dr. Yan's team also offers complimentary online video tutorials , facilitating a swift mastery of DPABINet. This freely accessible, open-source toolbox is tailored to support both newcomers and seasoned users, catalyzing the deployment of brain network research methodologies in both academic and clinical translational studies.

DPABINet marks a significant leap forward in brain imaging data analysis, furnishing researchers with a potent, user-friendly instrument to delve into the complexities of brain networks and their implications in health and disease. This platform enhances the efficiency and precision of constructing and analyzing brain networks, propelling brain science research forward and offering novel investigative tools for the diagnosis and treatment of brain function disorders.

The introduction and utilization of DPABINet promise to significantly advance the popularity and application of brain network research techniques, accelerating progress in brain functional and structural imaging studies, and providing robust support for clinical translational research endeavors.

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