Opuscula Mathematica
Opuscula Math. 37, no. 4 (), 617-639
Opuscula Mathematica

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.
Keywords: Fan's condition, circumference, hamiltonian cycle, pancyclicity.
Mathematics Subject Classification: 05C38, 05C45.
Cite this article as:
Wojciech Wideł, Fan's condition on induced subgraphs for circumference and pancyclicity, Opuscula Math. 37, no. 4 (2017), 617-639, http://dx.doi.org/10.7494/OpMath.2017.37.4.617
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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