Cycles of each even lengths in balanced bipartite digraphs

TL;DR

This paper investigates the existence of cycles of various even lengths in strongly connected balanced bipartite digraphs under certain degree conditions, extending understanding of cycle structures in such graphs.

Contribution

It establishes new degree-based conditions that guarantee the presence of all even cycles up to a certain length in balanced bipartite digraphs.

Findings

Existence of a cycle of length 2a-2 or the digraph being a directed cycle.

Presence of all even cycles of length 2k for 1 ≤ k ≤ a-1 under certain conditions.

Almost all such digraphs contain all even cycles up to length 2a, except one specific exceptional case.

Abstract

Let $D$ be a strongly connected balanced bipartite directed graph of order $2a\geq 4$. Let $x,y$ be distinct vertices in $D$. $\{x,y\}$ dominates a vertex $z$ if $x\rightarrow z$ and $y\rightarrow z$; in this case, we call the pair $\{x,y\}$ dominating. In this paper we prove: (i). If $a\geq 4$ and $ max\{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, $D$ then contains a cycle of length $2a-2$ or $D$ is a directed cycle. (ii). If $D$ contains a cycle of length $2a-2\geq 6$ and $max \{d(x), d(y)\}\geq 2a-2$ for every dominating pair of vertices $\{x,y\}$, then for any $k$, $1\leq k\leq a-1$, $D$ contains a cycle of length $2k$. (iii). If $a\geq 4$ and $ max\{d(x), d(y)\}\geq 2a-1$ for every dominating pair of vertices $\{x,y\}$, then for every $k$, $1\leq k\leq a$, $D$ contains a cycle of length $2k$ unless $D$ is isomorphic to only one exceptional digraph of order eight.