TITLE: Even cycles in a graph
QUESTION [10 upvotes]: I'm at a dead end with the following question,
It seems simple enough but I cant seem to see it.
Let G be a simple undirected graph with minimal degree 3, show that G contains an even cycle.
Thanks.

REPLY [11 votes]: Consider the maximal path $P$ in the graph G. Let it be $u_1u_2...u_k$. By the maximality of the path $P$, we know that all neighbors of $u_1$ belong to {$u_2,u_3,...,u_k$}. Since $3\leq \deg(u_1)$, let $u_i,u_j$ be two neighbors of $u_1$ such that $2<i<j$. Now it is easy to show that one of the following cycles must be even:
$$u_1u_2..u_iu_1$$
$$u_1u_2..u_ju_1$$
$$u_1u_iu_{i+1}..u_{j-1}u_ju_1$$
This is becasue the sum of lengths of all three cycles $=(i)+(j)+(j-i)=2j$ is even, thus it is impossible that the three lengths are odd.