TY  - JOUR
ID  - Wideł2017
LB  - Wideł2017
AU  - Wideł, Wojciech
TI  - Fan's condition on induced subgraphs for circumference and pancyclicity
ST  - Fan's condition on induced subgraphs for circumference and pancyclicity
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2017
VL  - 37
IS  - 4
SP  - 617
EP  - 639
DO  - http://dx.doi.org/10.7494/OpMath.2017.37.4.617
UR  - http://dx.doi.org/10.7494/OpMath.2017.37.4.617
AB  - 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.
KW  - Fan's condition
KW  - circumference
KW  - hamiltonian cycle
KW  - pancyclicity
SN  - 1232-9274
ER  -