Question

# Let $G$ be a graph. The line graph of $G$ is a new graph $L(G)$ whose vertices are the edges of $G$; two vertices of $L(G)$ are adjacent if, as edges of $G$, they share a common end point. In symbols:$V[L(G)]=E(G)\text{ and }E[L(G)]=\{e_1e_2:|e_1\cap e_2|=2\}$Prove or disprove the following statements about the relationship between a graph and its line graph : a. If $G$ is Eulerian, then $L(G)$ is also Eulerian. b. If $G$ has a Hamiltonian cycle, then $L(G)$ is Eulerian. (See Exercise 50.16 for the definition of a Hamiltonian cycle.) c. If $L(G)$ is Eulerian, then $G$ is also Eulerian. d. If $L(G)$ is Eulerian, then $G$ has a Hamiltonian cycle.

Solution

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DEFINITIONS

An Eulerian trial is a walk in a graph that contains all edges in the graph exactly once.

An Eulerian tour is an Eulerian trial that beings and ends at the same vertex.

A graph is $\textbf{Eulerian}$ if $G$ contains an Eulerian tour.

A $\textbf{Hamilton cycle}$ is a cycle that passes through every vertex exactly once.

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