Prove that if $G$ contain an Eulerian circuit, then its line graph $L(G)$ also contains an Eulerian circuit.


Assume that the graph as undirected graph

then the vertice are $1,2,3,4,5,6,7$

Let $G$ be the graph with edge $(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,1)$

Apparently graph and its line graph that both will contain Eulerian circuits.

But I don't have any idea to prove it, Could you guys to help me ?


The proof is pretty easy:

Theorem: A connected graph $G$ has Euler cycle if and only if every vertex of $G$ is of even degree.

It easy to see that if all vertices of $G$ has even degree, then all the vertices of $L(G)$ has even degree (since each edge in $G$ is connected to odd number of edges from each of its two ends"vertices", and the sum of two odd numbers is even: $(2p+1) + (2q+1) = 2(p+q+1)$).

Now using the above Theorem we have the result for $L(G)$.


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