Section 2.3, #8
March 29, 2012
Problem 8
(a) Show that for every connected graph G there is a spanning tree T of
G such that diam(T ) 2diam(G).
(b) Prove or disprove: For every positive integer k , there is a connected
graph G and a spanning tree T of G
Section 3.1, #10
May 4, 2012
Problem 10
(a) Prove that every Eulerian graph of odd order has three vertices of the same
degree.
(b) Prove that for each odd integer n 3, there exists exactly one Eulerian
graph of order n containing exactly three vertices o
Section 1.1, #4
January 31, 2012
Problem 4
A graph G has order n = 3k + 3 for some positive integer k . Every vertex of G
has degree k + 1, k + 2, or k + 3. Prove that G has at least k + 3 vertices of
degree k + 1, or at least k + 2 vertices of degree k +
Section 1.1, #16
January 31, 2012
Problem 16
Let G1 and G2 be self-complementary graphs, where G2 has even order n. Let
G be the graph obtained from G1 and G2 by joining each vertex of G2 whose
degree is less than n/2 to every vertex of G1 . Show that G i
Section 1.2, #5
February 23, 2012
Problem 5
Let s : d1 , d2 , . . . , dn be a graphical sequence with = d1 d2 dn .
Show, for each integer k with 1 k n, that there exists a graph G with
V (G) = cfw_v1 , v2 , . . . vn where deg vi = di for 1 i n having the
Section 1.3, #24
February 23, 2012
Problem 24
Prove that a nontrivial graph G is bipartite if and only if G contains no induced
odd cycle.
Solution
We already know that G is bipartite if and only if G contains no odd cycle. We
will show that G contains no
Section 1.3, #30
February 24, 2012
Problem 30
Every complete graph is the periphery of itself. Can a complete graph be the
periphery of a connected graph G with diam(G) 2?
Solution
No. If v Per(G) then there is some u V (G) with d(u, v ) = e(v ) = diam(G)
Section 2.1, #5
March 2, 2012
Problem 5
Prove that if v is a cut-vertex of a connected graph G, then v is not a cut-vertex
of G.
Solution
Suppose v is a cut-vertex in G. Then G v is disconnected. Let u, w
V (G v ) = V (G v ).
If u and w are in dierent co
Section 2.2, #20
March 6, 2012
Problem 20
Prove that if T is a tree of order n 2 that is not a star, then T is isomorphic
to a subgraph of T .
Solution
We will prove this by induction on n, the order of T .
If n = 2, 3, any tree is a star. We begin with n
Section 2.3, #26
March 29, 2012
Problem 26
Let G be a connected weighted graph. Prove that every minimum spanning tree
of G can be obtained by Kruskals algorithm.
Solution
Suppose T is a minimum spanning tree for G. Label the edges of T , e1 , e2 , e3 , .
Section 2.4, #18
March 29, 2012
Problem 18
Let G1 and G2 be two k -connected graphs, where k 2, and let G be the
set of all graphs obtained by adding k edges between G1 and G2 . Determine
maxcfw_(G) : G G.
Solution
We rst observe that for any G G , (G) k
Section 6.1, #4
May 4, 2012
Problem 4
Prove that there exists only one 4-regular maximal planar graph.
Solution
We rst check that there is a 4-regular maximal planar graph. Let G be the
graph depicted below, with degree sequence 4, 4, 4, 4, 4, 4. Clearly
MATH 4171 Graph Theory Homework Set VII - Solutions
1. Prove without using Theorem 8.24 that every planar graph is 6-colorable.
SPRING 2006
Proof. Let G be a planar simple graph. Then every induced subgraph G of G is also a planar simple graph. By
MATH 4171 Graph Theory
SPRING 2006
Homework Set VII (4 problems) Due date : Monday 5-1-06
1. Prove without using Theorem 8.24 that every planar graph is 6-colorable. 2. For the graph G below, prove that (G) = 4 by showing: (a) (G) 4; and (b) (G)
MATH 4171 Graph Theory Homework Set VI -Solutions
SPRING 2006
1. Let G = (V, E) be a plane graph of minimum degree at least three. Prove that G has a region of size at most ve. Proof. Let d1 , d2 , ., d|V | be the degrees of G. Then 2|E| = d1 + d2
MATH 4171 Graph Theory
SPRING 2006
Homework Set I (Four problems) Due date : Monday 1-30-06
1. Describe a real world problem (preferable from your own eld) that can be modeled by graphs. Specify vertices, edges, and the incidence relation for your
MATH 4171 Graph Theory Homework Set I - Solutions
SPRING 2006
1. Describe a real world problem (preferable from your own eld) that can be modeled by graphs. Specify vertices, edges, and the incidence relation for your graph. . 2. Is there a simple
MATH 4171 Graph Theory Homework Set II (5 problems) Due date : Monday 2-13-06
SPRING 2006
1. For the graph below, nd the distance from u to h, and a shortest u-h path (with respect to the given numbers on each edge). Show your work
d 4 a 1 u 3 2 7
MATH 4171 Graph Theory Homework Set II -Solutions
SPRING 2006
1. For the graph below, nd the distance from u to h, and a shortest u-h path (with respect to the given numbers on each edge). Show your work.
d 4 a 1 u 3 2 7 b 2 6 e 3 1 9 c 1 7 5 5 1 2
MATH 4171 Graph Theory Homework Set III Due date : Monday 3-6-06
SPRING 2006
1. Consider the two arc-disjoint s-t dipaths, sabhgt and sef t, in the following digraph. (a) Find an augmenting path with respect to these two paths; (b) Using the given
MATH 4171 Graph Theory Homework Set III -Solutions
SPRING 2006
1. Consider the two arc-disjoint s-t dipaths, sabhgt and sef t, in the following digraph. (a) Find an augmenting path with respect to these two paths; (b) Using the given two paths and
MATH 4171 Graph Theory Homework Set IV (4 problems) Due date : Monday 3-20-06
SPRING 2006
1. Let G = (X, Y, E) be an r-regular (r > 0) bipartite graph. Prove that G has a perfect matching.
2. Find a maximum matching in the following graph. Prove t
MATH 4171 Graph Theory Homework Set IV -Solutions
SPRING 2006
1. Let G = (X, Y, E) be an r-regular (r > 0) bipartite graph. Prove that G has a perfect matching. Proof. Let V = X Y . We prove that 1 |M |
|V | . 2 |V | , 2
which means that G has
MATH 4171 Graph Theory Homework Set V (4 problems) Due date : Monday 4-3-06
SPRING 2006
1. Theorem 4.8 states that a necessary condition for a graph G = (V, E) to be Hamiltonian is that G S has at most |S| components, for all S V . Is this condit
MATH 4171 Graph Theory Homework Set V Solutions
SPRING 2006
1. Theorem 4.8 states that a necessary condition for a graph G = (V, E) to be Hamiltonian is that G S has at most |S| components, for all S V . Is this condition sucient? In other words
MATH 4171 Graph Theory Homework Set VI (3 problems) Due date : Monday 4-17-06
SPRING 2006
1. Let G be a plane graph of minimum degree at least three. Prove that G has a region of size at most ve.
2. Decide if the following graphs are planar. You n
MATH 4171
SPRING 2006
Graph Theory
A supplement
Guoli Ding http:/math.lsu.edu/ ding Last Updated: May 3, 2006
Copyright c 1996, by Guoli Ding. All Rights Reserved.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .