Disjoint cycles covering specified vertices in bipartite graphs with partial degrees

TL;DR

This paper proves conditions under which a balanced bipartite graph contains disjoint cycles covering a specified subset of vertices, with degree conditions and subset size bounds shown to be optimal.

Contribution

It establishes new extremal conditions for disjoint cycle coverings in bipartite graphs with partial degree constraints, extending previous results.

Findings

If |S| ≥ 2k+2, G contains k disjoint cycles covering S with each cycle having at least two vertices of S.

The degree condition and size bound for S are proven to be best possible.

For |S|=2k+1, G still contains k disjoint cycles covering S with each cycle containing at least two vertices of S.

Abstract

Let $k$ be a positive integer. Let $G$ be a balanced bipartite graph of order $2n$ with bipartition $(X, Y)$, and $S$ a subset of $X$. Suppose that every pair of nonadjacent vertices $(x,y)$ with $x\in S, y\in Y$ satisfies $d(x)+d(y)\geq n+1$. We show that if $|S|\geq 2k+2$, then $G$ contains $k$ disjoint cycles covering $S$ such that each of the $k$ cycles contains at least two vertices of $S$. Here, both the degree condition and the lower bound of $|S|$ are best possible. And we also show that if $|S|=2k+1$, then $G$ contains $k$ disjoint cycles such that each of the $k$ cycles contains at least two vertices of $S$.