Cycle in a graph

A cycle is a closed path: its final vertex is joined back to its initial vertex, and no other vertex is repeated.

The girth of a graph containing a cycle is the length of its shortest cycle. A forest is conventionally assigned infinite girth.

An odd cycle is a cycle in a graph with an odd number of edges.

The cycle graph $C_n$ has vertices $v_1,\dots ,v_n$ and edges $v_i{v}_{i+1}$, with indices read cyclically.

A wheel graph is the join of graphs of a cycle graph and a single vertex. Every wheel with a rim of at least three vertices has a $K_4$ graph minor: partition the rim into three nonempty consecutive connected sets and use its centre as the fourth branch set of a graph minor.

A Hamilton cycle is a cycle containing every vertex of its graph.

An $n$-vertex graph is pancyclic when it contains $C_{\ell }$ for every $3\le \ell \le n$.