Fan's condition on induced subgraphs for circumference and pancyclicity

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