Fan's condition on induced subgraphs for circumference and pancyclicity

Abstract

Let $\mathcal{H}$ be a family of simple graphs and $k$ be a positive integer. We say that a graph $G$ of order $n\geq k$ satisfies Fan's condition with respect to $\mathcal{H}$ with constant $k$, if for every induced subgraph $H$ of $G$ isomorphic to any of the graphs from $\mathcal{H}$ the following holds: $\forall u,v\in V(H)\colon d_H(u,v)=2,\Rightarrow \max{d_G(u),d_G(v)}\geq k/2.$ If $G$ satisfies the above condition, we write $G\in\mathcal{F}(\mathcal{H},k)$. In this paper we show that if $G$ is $2$-connected and $G\in\mathcal{F}({K_{1,3},P_4},k)$, then $G$ contains a cycle of length at least $k$, and that if $G\in\mathcal{F}({K_{1,3},P_4},n)$, then $G$ is pancyclic with some exceptions. As corollaries we obtain the previous results by Fan, Benhocine and Wojda, and Ning.