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Fan's condition on induced subgraphs for circumference and pancyclicity

Wideł W.

Opuscula Mathematica

2017

Vol. 37, no. 4

617--639

Let H be a family of simple graphs and k be a positive integer. We say that a graph G of order n ≥ k satisfies Fan's condition with respect to H with constant k, if for every induced subgraph H of G isomorphic to any of the graphs from H the following holds: [formula] If G satisfies the above condition, we write [formula]. In this paper we show that if G is 2-connected and [formula], then G contains a cycle of length at least k, and that if [formula], then G is pancyclic with some exceptions. As corollaries we obtain the previous results by Fan, Benhocine and Wojda, and Ning.

Edge condition for hamiltonicity in balanced tripartite graphs

Adamus J.

Opuscula Mathematica

2009

Vol. 29, no. 4

337-343

A well-known theorem of Entringer and Schmeichel asserts that a balanced bipartite graph of order 2n obtained from the complete balanced bipartite Kn,n by removing at most n - 2 edges, is bipancyclic. We prove an analogous result for balanced tripartite graphs: If G is a balanced tripartite graph of order 3n and size at least 3n(2) - 2n + 2, then G contains cycles of all lengths.