On the cycle structure of hamiltonian k-regular bipartite graphs of order 4k

TL;DR

This paper proves that certain Hamiltonian bipartite graphs with regular degree contain long cycles, and under specific conditions, are bipancyclic, extending understanding of cycle structures in these graphs.

Contribution

It establishes the existence of near-maximum length cycles and bipancyclicity in Hamiltonian bipartite regular graphs of order greater than 8.

Findings

Hamiltonian n/2-regular bipartite graphs of order 2n>8 contain a cycle of length 2n-2.

Such graphs are bipancyclic if a cycle omits a pair of adjacent vertices.

Results extend cycle structure knowledge in bipartite regular graphs.

Abstract

It is shown that a hamiltonian $n/2$-regular bipartite graph $G$ of order $2n>8$ contains a cycle of length $2n-2$. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then $G$ is bipancyclic.