A theorem on even pancyclic bipartite digraphs

TL;DR

This paper proves a new theorem establishing conditions under which strongly connected balanced bipartite digraphs contain cycles of all even lengths from 2 up to twice the size of the partite sets.

Contribution

It introduces a novel degree sum condition that guarantees the existence of all even cycles in such bipartite digraphs.

Findings

Strongly connected balanced bipartite digraphs with degree sum condition contain all even cycles.

The degree sum condition is sufficient for pancyclicity in bipartite digraphs.

The theorem extends previous results on cycle existence in bipartite graphs.

Abstract

We prove that a strongly connected balanced bipartite directed graph of order $2a\geq 6$ with partite sets $X$ and $Y$ contains cycles of every length $2, 4, \ldots , 2a$, provided $d(x)+d(y)\geq 3a$ for every pair of vertices $x$, $y$ either both in $X$ or both in $Y$.