If a graph $G$ has no isolated vertices and no even cycle, then every block of $G$ is an edge of cycle.
I am trying to understand the following proof of this fact:
- A block with 2 vertices is an edge. (Got it)
- A block $H$ with more than 2 vertices is 2-connected; and thus has an ear decomposition. (Got it)
- If $H$ is not a single cycle, then the addition of the first ear to the first cycle creates a subgraph in which a pair of vertices is connected by 3 pairwise internally-disjoint subgraphs. (Don't understand.)
- By the pigeonhole principle, two of the paths have length of same parity. (Don't understand. Why does the pigeonhole principle tell us that two of the paths have the same parity?)
- Thus their union is even. So H must be a single cycle. (I'll understand once I understand premises 3,4.)