,MAT3707 Assignment 3 (COMPLETE ANSWERS) 2024
(156556) - DUE 30 August 2024 ; 100% TRUSTED
Complete, trusted solutions and explanations.
QUESTION 1 Consider the following two graphs: a b d c e f G1
u v w y x z G2 Are G1 and G2 isomorphic? A. Yes, because
they both have the same degree sequence. B. Yes, because
function g with g(a) = u, g(b) = v, g(c) = w, g(d) = x, g(e) = y,
and g(f) = z is an isomorphism. C. Yes, because function h with
h(a) = z, h(b) = w, h(c) = x, h(d) = y, h(e) = u, and h(f) = v is an
isomorphism. D. Yes, because they both have the same number
of vertices, the same number of edges and are both connected. E.
No, they are not isomorphic. QUESTION 2 Which of the
following is a necessary condition for a graph to be bipartite? A.
The graph has no odd cycles. B. The graph has even cycles. C.
The graph has an Euler cycle. D. The graph is planar. E. The
graph has a Hamiltonian circuit. QUESTION 3 In a connected
graph, if every vertex has an even degree, what can you
conclude? A. The graph has an Euler cycle. B. The graph is
planar. C. The graph has a Hamiltonian circuit. D. The graph is
bipartite. E. The graph is a tree. QUESTION 4 Which of the
following is NOT a property of a Hamiltonian circuit in a graph?
Page 2 A. It visits every vertex exactly once. B. It forms a cycle.
C. It forms a path. D. It must contain exactly two edges incident
to each vertex. E. It may contain repeated vertices. QUESTION
5 Consider the following graph: v2 v4 v5 v1 v3 Does this graph
have an Euler cycle? A. No, because there is no edge between
v1 and v5. B. No, because v3 does not have degree 2. C. No,
because it is not a complete graph. D. Yes, but not all edges
, incident to v3 form part of the Euler cycle. E. Yes, and it will
visit v3 more than once QUESTION 6 For the graph in Question
5, what is its chromatic number? A. 1 B. 2 C. 3 D. 4 E. 5
QUESTION 7 What is the chromatic number of a complete
graph with n vertices? A. n − 1 B. n C. n 2 D. (n)(n−1) 2 E.
(n)(n+1) 2 QUESTION 8 In the complete bipartite graph K4,3,
what is the maximum possible chromatic number? A. 7 B. 6 C. 4
D. 3 E. 2 Page 3 QUESTION 9 Which of the following graphs
with n vertices, n ≥ 5 has the highest upper bound for its
chromatic number? A. Complete graph Kn B. Bipartite graph C.
Tree D. Planar graph E. Disconnected graph QUESTION 10 A
complete n-ary tree is a tree in which each vertex has n children
or no children. Suppose one such tree has 10 internal vertices
and 41 leaves. What is the value of n? A. 2 B. 3 C. 4 D. 5 E. 6
QUESTION 11 Consider the following graph: v2 v4 v5 v1 v3
Does this graph have a Hamiltonian circuit? A. No, because it is
a complete graph. B. No, because not every vertex has even
degree. C. No, because every vertex has even degree. D. Yes,
because it is a complete graph. E. Yes, because every vertex has
even degree. QUESTION 12 In a graph with n vertices, what is
the maximum number of edges it can have without forcing it to
contain a Hamiltonian cycle? A. n B. n − 1 C. n 2 D. (n−1)(n−2)
2 + 1 E. n(n−1) 2 QUESTION 13 What is the chromatic number
of an unrooted tree with n vertices? Page 4 A. n B. 2 C. 1 D. n −
1 E. n−1 2 QUESTION 14 To avoid forming a cycle, a
connected graph with 7 vertices can have at most how many
edges? A. 9 B. 8 C. 7 D. 6 E. 5 QUESTION 15 How many
different trees are there with four labelled vertices A, B, C and
D? A. 16 B. 32 C. 44 D. 64 E. 96 QUESTION 16 In a graph
