In graph $G$, for each vertex v with even and positive degree, There is a non-trivial cycle $(\ge 3)$ from $v$ to $v$.

Question

Having troubles with the following problem, thinking about induction, but can't go further:

In graph $G$, for each vertex $v$ with even and positive degree, There is a non-trivial cycle $(\ge 3)$ from $v$ to $v$.

Answer 1

Actually with these information, statement doesn't imply Euler circuit exists. Notice that we are not given $G$ is connected or finite. But still, since $G$ is not a multigraph by convention, smallest component of $G$ (may be connected or disconnected) satisfying the condition given is $K_3$, which has a cycle of length at least $3$ for all of its vertices. For the rest, you can use the argument "for every vertex in $G$, number of enters to that vertex is equal to number of exists from that vertex". If $G$ is a finite graph, you can eventually find a cycle of length greater than or equal to $3$ for every vertex. If $G$ is not finite, I'm not sure infinite cycles are included to cycles of length more than $3$ or not.