b What is the minimum possible number of vertices of degree 1 for a tree on five vertices?
c What is the maximum possible vertex degree for a tree on 5 vertices? d List the possible sets of vertex degrees for trees on 5 vertices. 6
A simple, connected Eulerian graph has 5 vertices and 7 edges. Deduce the degrees of the vertices.
7
This question is about the seven graphs (A) K , (B) K , (C) K , (D) K and (G ) K . 3
4
5
2,3
, (E) K
2,4
, (F ) K
3,3
3,4
PL E
a How many edges does each of these graphs have? b Which of the six graphs are Eulerian? 8
a Give an example of a graph that is Hamiltonian but not Eulerian. b Give an example of a graph that is Eulerian but not Hamiltonian.
9 10
There are 6 different trees with 6 vertices. Draw an example of each and state its degrees.
a Are complete graphs Hamiltonian?
How many different trees on:
a
4
vertices
b
5
vertices
SA
11
M
b Are complete bipartite graphs Hamiltonian?
c
12
6
vertices can be made as subgraphs of K
3,3
?
A graph has 4 vertices. The vertex degrees are 1, 2, 3, 4.
a How many edges does the graph have? b Explain how the vertex degrees show that the graph is not simple-connected. c How many different graphs fit this description?
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