@Article{Wideł2017,
author = "Wideł, Wojciech",
title = "Fan's condition on induced subgraphs for circumference and pancyclicity",
journal = "Opuscula Math.",
year = "2017",
volume = "37",
number = "4",
pages = "617--639",
doi = "http://dx.doi.org/10.7494/OpMath.2017.37.4.617",
url = "http://dx.doi.org/10.7494/OpMath.2017.37.4.617",
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.",
keywords = "Fan's condition, circumference, hamiltonian cycle, pancyclicity",
issn = "1232-9274"
}