(d) State Euler's Theorem on Eulerian circuits in multigraphs. (e) Use Euler's Theorem to decide if the multigraph below has an Eulerian circuit.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 80EQ
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d,e

4. Let G be a graph with n vertices.
(a) State what it means for G to be a connected graph.
(b) Suppose that G is connected. Find a lower bound on |E(G)| (the number of edges of G)
and justify your answer.
(c) Suppose that G has a Hamilton cycle and that v € V(G) is an arbitrary vertex of G.
Prove that G v is connected.
(d) State Euler's Theorem on Eulerian circuits in multigraphs.
(e) Use Euler's Theorem to decide if the multigraph below has an Eulerian circuit.
Transcribed Image Text:4. Let G be a graph with n vertices. (a) State what it means for G to be a connected graph. (b) Suppose that G is connected. Find a lower bound on |E(G)| (the number of edges of G) and justify your answer. (c) Suppose that G has a Hamilton cycle and that v € V(G) is an arbitrary vertex of G. Prove that G v is connected. (d) State Euler's Theorem on Eulerian circuits in multigraphs. (e) Use Euler's Theorem to decide if the multigraph below has an Eulerian circuit.
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