Variations of Pancyclic Graphs

Abstract

A graph \( G \) of order \( n \) is pancyclic if it contains a cycle of length \( \ell \) for every \( \ell \) such that \( 3 \leq \ell \leq n \). If the graph is bipartite, then it contains no cycles of odd length. A balanced bipartite graph \( G \) of order \( 2n \) is bipancyclic if it contains a cycle of length \( \ell \) for every even \( \ell \), such that \( 4 \leq \ell \leq 2n \). A graph \( G \) of order \( n \) is called \( k \)-semipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the cycle lengths in \( G \), i.e., for no \( \ell \leq n-k \) is it the case that each of \( C_\ell, \ldots, C_{\ell+k} \) is missing from \( G \). Generalizing this to bipartite graphs, a bipartite graph \( G \) of order \( n \) is called \( k \)-semibipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the even cycle lengths in \( G \), i.e., for no \( \ell \leq n-2k \) is it the case that each of \( C_{2\ell}, \ldots, C_{2\ell+2k} \) is missing from \( G \).

In this paper we generalize a result of Hakimi and Schmeichel in several ways. First to \( k \)-semipancyclic, then to bipartite graphs, giving a condition for a hamiltonian bipartite graph to be bipancyclic or one of two exceptional graphs. Finally, we give a condition for a hamiltonian bipartite graph to be \( k \)-semibipancyclic or a member of a very special class of hamiltonian bipartite graphs.