Forbidden Subgraphs and Cycle Extendability

Abstract

A graph $G$ on $n$ vertices is $pancyclic$ if $G$ contains cycles of all lengths $\ell$ for $3\le \ell \le n$ and $G$ is $cycleextendable$ if for every non-hamiltonian cycle $C\subset G$ there is a cycle $C^{\prime }\subset G$ such that $V(C)\subset V(C^{\prime })$ and $|V(C^{\prime })\setminus V(C)|=1$. We prove that

1. every $2$-connected $K_{1,3}$-free graph is pancyclic, if $G$ is $P_5$-free and $n\ge 6$, if $G$ is $P_6$-free and $n\ge 10$, or if $G$ is $P_7$-free, deer-free and $n\ge 14$, and
2. every $2$-connected $K_{1,3}$-free and $Z_2$-free graph on $n\ge 10$ vertices is cycle extendible using at most two chords of the cycle.