8/19/2019 tut161
1/2
DEPARTMENT OF MATHEMATICSII Semester 2015-2016
MTL 768/MAL 656/MAL 468 Graph TheoryTutorial
Sheet 1
1. Prove that every graph of order n ≥ 2 has at
least two vertices of equal degree.
2. What is the maximum number of edges a graph of order
n can have?
3. Let p be an integer such that 0 ≤ p
≤ n. Does there exist a graph having exactly p
oddvertices?
4. A graph of order 20 and size 50 has 12 vertices of degree
a and remaining vertices of degree b. Find
the possible values of a and b.
5. Prove that either G or its complement is
connected.
6. Let G be a graph containing p vertices of
degree p and q vertices of
degree q and
p+q = n.If G contains an odd
vertex, then prove that every vertex of G is
odd.
7. For k ≥ 2, prove that a k-regular
bipartite graph has no cut edge.
8. Let G be a graph with at least two vertices. Prove
or disprove:
a. Deleting a vertex of degree ∆(G) cannot increase the average
degree.
b. Deleting a vertex of degree δ (G) cannot reduce
the average degree.
9. Prove that every graph with n vertices and
n edges contains a cycle.
10. Let p1, p2, . . . , pn be n points in
the plane such that the distance between any two points isat least
one. Let G = (V, E ) be the graph such that
V = { p1, p2, . . . , pn} and
E = { pi p j|distance between
pi and p j is exactly one}. Show that
∆(G) = 6.
11. If G has δ (G) ≥ 2,
then show that G contains a cycle of length at least
δ (G) + 1.
12. . If G has n vertices and
δ (G) ≥ (n− 1)/2 , then prove that
G is connected.
13. Let G be a simple graph having no isolated vertex
and no induced subgraph with exactly
two edges. Prove that G is a complete graph.
14. Prove or disprove: If u and v
are the only vertices of odd degree in a graph G, then
Gcontains a u − v path.
15. . Let G be a simple graph with adjacency matrix
A and incidence matrix M . Provethat the
degree of vi is the i
th diagonal entry in A2 and in MM T . What
do the entriesin position (i, j) of A2 and
MM T say about G?
16. Prove that a self complementary graph with n
vertices exists iff n or n − 1 is
divisibleby 4.
1
8/19/2019 tut161
2/2
17. Let G be a bipartite graph with
n vertices and m edges. Show that
m ≤ n2/4.
18. Show that a connected graph with n vertices has
at least n− 1 edges.
19. Let S be a set of six persons. Any two
persons in S are either friends or strangers.
Provethat S contains three persons who are mutual
friends or mutual strangers.
20. Let P 1 and P 2 be two
distinct paths from x to y in a graph G.
show that P 1∪P 2 containsa cycle. (Prove it by
construction).
21. Prove that G is connected if and only if for
every partition of its vertices into twononempty sets, there is an
edge with end points in both sets.
22. Let p and q be nonnegative
integers, not both equal to 0. Show that there is not alwaysa graph
having p odd vertices and q even
vertices. However, show that there is such agraph
if p is required to be even.
23. Suppose G is a graph of order 3n with
D(G) {n, n + 1, n + 2}. Show that G
containseither (i) at least n vertices of degree
n, (ii) at least n + 2 vertices of degree
n + 1, or(iii) at least n + 1 vertices of
degree n + 2.
24. If an n-vertex simple graph has n −
1 distinct vertex degrees, then which degree isrepeated?
2
