Introductory Combinatorics

Richard A. Brualdi

Chapter 12

More on Graph Theory - all with Video Answers

Educators

Chapter Questions

Problem 1

Prove that isomorphic graphs have the same chromatic number and the same chromatic polynomial.

Chris Trentman

Chris Trentman

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Problem 2

Prove that the chromatic number of a disconnected graph is the largest of the chromatic numbers of its connected components.

Nick Johnson

Numerade Educator

Problem 3

Prove that the chromatic polynomial of a disconnected graph equals the product of the chromatic polynomials of its connected components.

Raj Bala

Numerade Educator

Problem 4

Prove that the chromatic number of a cycle graph $C_{n}$ of odd length equals $3 .$

Nick Johnson

Numerade Educator

Problem 5

Determine the chromatic numbers of the following graphs:

Nick Johnson

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Problem 6

Prove that a graph with chromatic number equal to $k$ has at least $\left(\begin{array}{c}k \\ 2\end{array}\right)$ edges.

Nick Johnson

Numerade Educator

Problem 7

Prove that the greedy algorithm always produces a coloring of the vertices of $K_{m, n}$ in two colors $(m, n \geq 1)$.

Chris Trentman

Numerade Educator

Problem 8

Let $G$ be a graph of order $n \geq 1$ with chromatic polynomial $p_{G}(k)$.
(a) Prove that the constant term of $p_{G}(k)$ equals $0 .$
(b) Prove that the coefficient of $k$ in $p_{G}(k)$ is nonzero if and only if $G$ is connected.
(c) Prove that the coefficient of $k^{n-1}$ in $p_{G}(k)$ equals $-m$, where $m$ is the number of edges of $G$.

WZ

Wen Zheng

Numerade Educator

Problem 9

Let $G$ be a graph of order $n$ whose chromatic polynomial is $p_{G}(k)=k(k-1)^{n-1}$ (i.e., the chromatic polynomial of $G$ is the same as that of a tree of order $n$ ). Prove that $G$ is a tree.

Vikash Ranjan

Numerade Educator

Problem 10

What is the chromatic number of the graph obtained from $K_{n}$ by removing one edge?

Carson Merrill

Numerade Educator

Problem 11

Prove that the chromatic polynomial of the graph obtained from $K_{n}$ by removing an edge equals
$$
[k]_{n}+[k]_{n-1}
$$

Carson Merrill

Numerade Educator

Problem 12

What is the chromatic number of the graph obtained from $K_{n}$ by removing two edges with a common vertex?

Carson Merrill

Numerade Educator

Problem 13

What is the chromatic number of the graph obtained from $K_{n}$ by removing two edges without a common vertex?

Carson Merrill

Numerade Educator

Problem 14

Prove that the chromatic polynomial of a cycle graph $C_{n}$ equals
$$
(k-1)^{n}+(-1)^{n}(k-1)
$$

Wendi Zhao

Numerade Educator

Problem 15

Prove that the chromatic number of a graph that has exactly one cycle of odd length is 3 .

Nick Johnson

Numerade Educator

Problem 16

Prove that the polynomial $k^{4}-4 k^{3}+3 k^{2}$ is not the chromatic polynomial of any graph.

Ekaveera Kumar

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Problem 17

Use Theorem $12.1 .10$ to determine the chromatic number of the following graph:

Nick Johnson

Numerade Educator

Problem 18

Use the algorithm for computing the chromatic polynomial of a graph to da termine the chromatic polynomial of the graph $Q_{3}$ of vertices and edges of in three-dimensional cube.

Nick Johnson

Numerade Educator

Problem 19

Find a planar graph that has two different planar representations such that, for some integer $f$, one has a region bounded by $f$ edge-curves and the other has tu. such region.

Chris Trentman

Numerade Educator

Problem 20

Give an example of a planar graph with chromatic number 4 that does nol contain a $K_{4}$ as an induced subgraph.

Nick Johnson

Numerade Educator

Problem 21

A plane is divided into regions by a finite number of straight lines. Prove thal the regions can be colored with two colors in such a way that regions which share a boundary are colored differently.

Prathan Jarupoonphol

Prathan Jarupoonphol

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Problem 22

Repeat Exercise 21, with circles replacing straight lines.

Linda Hand

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Problem 23

Let $G$ be a connected planar graph of order $n$ having $e=3 n-6$ edges. Provr. that, in any planar representation of $G$, each region is bounded by exactly i edge-curves.

Chris Trentman

Numerade Educator

Problem 24

Prove that a connected graph can always be contracted to a single vertex.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 25

Verify that a contraction of a planar graph is planar.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 26

Let $G$ be a planar graph of order $n$ in which every vertex has the same degree
k. Prove that $k \leq 5$.

Chris Trentman

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Problem 27

Let $G$ be a planar graph of order $n \geq 2$. Prove that $G$ has at least two vertices whose degrees are at most $5 .$

Chris Trentman

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Problem 28

A graph is called color-critical provided each subgraph obtained by removing a vertex has a smaller chromatic number. Let $G=(V, E)$ be a color-critical graph. Prove the following:
(a) $\chi\left(G_{V-\{x\}}\right)=\chi(G)-1$ for every vertex $x$.
(b) $G$ is connected.
(c) Each vertex of $G$ has degree at least equal to $\chi(G)-1$.
(d) $G$ does not have an articulation set $U$ such that $G_{U}$ is a complete graph.
(e) Every graph $H$ has an induced subgraph $G$ such that $\chi(G)=\chi(H)$ and $G$ is color-critical.

Nick Johnson

Numerade Educator

Problem 29

Let $p \geq 3$ be an integer. Prove that a graph, each of whose vertices has degree at least $p-1$, contains a cycle of length greater than or equal to $p$. Then use Exercise 28 to show that a graph with chromatic number equal to $p$ contains a cycle of length at least $p$.

Nick Johnson

Numerade Educator

Problem 30

Let $G$ be a graph without any articulation vertices such that each vertex has degree at least 3. Prove that $G$ contains a subgraph that can be contracted to a $K_{4}$. (Hint: Begin with a cycle of largest length $p .$ By Exercise 29, we have $p \geq 4$. Now use Exercise 28 to obtain a proof of Hadwiger's conjecture for $p=4$.)

Chris Trentman

Numerade Educator

Problem 31

Let $G$ be a connected graph. Let $T$ be a spanning tree of $G$. Prove that $T$ contains a spanning subgraph $T^{\prime}$ such that, for each vertex $v$, the degree of $v$ in $G$ and the degree of $v$ in $T^{\prime}$ are equal modulo $2 .$

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 32

Find a solution to the problem of the 8 queens that is different from that given in Figure $12.9 .$

Morgan Cheatham

Morgan Cheatham

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Problem 33

Prove that the independence number of a tree of order $n$ is at least $\lceil n / 2]$.

Chris Trentman

Numerade Educator

Problem 34

Prove that the complement of a disconnected graph is connected.

WZ

Wen Zheng

Numerade Educator

Problem 35

Let $H$ be a spanning subgraph of a graph $G$. Prove that $\operatorname{dom}(G) \leq \operatorname{dom}(H)$.

Chris Trentman

Numerade Educator

Problem 36

For each integer $n \geq 2$, determine a tree of order $n$ whose domination number equals $\lfloor n / 2]$.

Anurag Kumar

Numerade Educator

Problem 37

Determine the domination number of the graph $Q_{3}$ of vertices and edges of a three-dimensional cube.

Norman Atentar

Numerade Educator

Problem 38

Determine the domination number of a cycle graph $C_{n}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 39

For $n=5$ and 6 , show that the domination number of the queens graph of an $n$ -by- $n$ chessboard is at most 3 by finding three squares on which to place queens so that every other square is attacked by at least one of the queens.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 40

Show that the domination number of the queens graph of a 7 -by-7 chessboard is at most 4 .

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 41

Show that the domination number of the queens graph of an 8 -by-8 chessboary is at most 5 .

Sheryl Ezze

Numerade Educator

Problem 42

Prove that an induced subgraph of an interval graph is an interval graph.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 43

Prove that an induced subgraph of a chordal graph is chordal.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 44

Prove that the only connected bipartite graphs that are chordal are trees.

Victoria Dollar

Victoria Dollar

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Problem 45

Prove that all bipartite graphs are perfect.

WZ

Wen Zheng

Numerade Educator

Problem 46

Let $G$ be a graph such that either $G$ or its complement $\bar{G}$ has an induced sub graph equal to a chordless cycle of odd length greater than 3. Prove that $G$ is not perfect.

Chris Trentman

Numerade Educator

Problem 47

Let $k$ be a positive integer, and let $G$ be a bipartite graph in which every vertex has degree $k$.
(a) Prove that $G$ has a perfect matching.
(b) Prove that the edges of $G$ can be partitioned into $k$ perfect matchings.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 48

Consider the graph $Q_{n}$ of vertices and edges of the $n$ -dimensional cube. Usiny. induction,
(a) Prove that $Q_{n}$ has a perfect matching for each $n \geq 1$.
(b) Prove that $Q_{n}$ has at least $2^{2^{n-2}}$ perfect matchings.

Prathan Jarupoonphol

Prathan Jarupoonphol

Numerade Educator

Problem 49

Prove that if a tree has a perfect matching, then it has exactly one perfer-1 matching.

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 50

Use Theorem $12.5 .4$ to prove the following theorem of Petersen (1891): A graph with every vertex of degree 3 and edge-connectivity at least 2 has a perfw-1 matching.

WZ

Wen Zheng

Numerade Educator

Problem 51

The Petersen graph $\mathcal{P}$ is the graph whose vertices are the ten 2 -subsets ul $\{1,2,3,4,5\}$ in which two vertices are joined by an edge if and only if thein 2-subsetss are disjoint.
(a) Draw a picture of the Petersen graph. (It can be drawn as a pentagon with a disjoint pentagram inside it-so 10 vertices and 10 edges - where there are an additional five edges joining each vertex of the pentagon to the corresponding vertex of the pentagram.)
(b) Verify that for each pair of vertices of $\mathcal{P}$ that are not joined by an edge, there is exactly one vertex joined by an edge to both.
(c) Verify that the smallest length of a cycle of $\mathcal{P}$ is 5 .

WZ

Wen Zheng

Numerade Educator

Problem 52

Prove that the edge-connectivity of $K_{n}$ equals $n-1$.

WZ

Wen Zheng

Numerade Educator

Problem 53

Give an example of a graph $G$ different from a complete graph for which $\kappa(G)=$ $\lambda(G) .$

AG

Ankit Gupta

Numerade Educator

Problem 54

Give an example of a graph $G$ for which $\kappa(G)<\lambda(G)$.

WZ

Wen Zheng

Numerade Educator

Problem 55

Give an example of a graph $G$ for which $\kappa(G)<\lambda(G)<\delta(G)$.

WZ

Wen Zheng

Numerade Educator

Problem 56

Determine the edge-connectivity of the complete bipartite graphs $K_{m, n} .$

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 57

Let $G$ be a graph of order $n$ with vertex degrees $d_{1}, d_{2}, \ldots, d_{n}$. Assume that the degrees have been arranged so that $d_{1} \leq d_{2} \leq \cdots \leq d_{n}$. Prove that, if $d_{k} \geq k$ for all $k \leq n-d_{n}-1$, then $G$ is a connected graph.

WZ

Wen Zheng

Numerade Educator

Problem 58

Let $G$ be a graph of order $n$ in which every vertex has degree equal to $d$.
(a) How large must $d$ be in order to guarantee that $G$ is connected?
(b) How large must $d$ be in order to guarantee that $G$ is 2-connected?

WM

William Mead

Numerade Educator

Problem 59

Determine the blocks of the graph given in Figure $12.12 .$

James Kiss

Numerade Educator

Problem 60

Prove that the blocks of a tree are all $K_{2}$ 's.

Robin Corrigan

Numerade Educator

Problem 61

Let $G$ be a connected graph. Prove that an edge of $G$ is a bridge if and only if it is the edge of a block equal to a $K_{2}$.

Brian Lin

Numerade Educator

Problem 62

Let $G$ be a graph. Prove that $G$ is 2 -connected if and only if, for each vertex $x$ and each edge $\alpha$, there is a cycle that contains both the vertex $x$ and the edge $\alpha$.

WZ

Wen Zheng

Numerade Educator

Problem 63

Let $G$ be a graph each of whose vertices has positive degree. Prove that $G$ is 2 . connected if and only if, for each pair of edges $\alpha_{1}, \alpha_{2}$, there is a cycle containing both $\alpha_{1}$ and $\alpha_{2}$.

Brian Lin

Numerade Educator

Problem 64

Prove that a connected graph of order $n \geq 2$ has at least two vertices that are not articulation vertices. (Hint: Take the two end vertices of a longest path.

Brian Lin

Numerade Educator