Connected $k$-factors in bipartite graphs

TL;DR

This paper proves that sufficiently connected bipartite graphs avoiding a specific subgraph structure always contain a connected k-factor, extending understanding of graph factors under forbidden subgraph conditions.

Contribution

It establishes the existence of a universal constant ensuring connected k-factors in connected, S_{k,l}-free bipartite graphs with high minimum degree.

Findings

Existence of a constant c(k,l) for connected k-factors

Connected S_{k,l}-free bipartite graphs with high minimum degree contain k-factors

Extension of factor existence results under forbidden subgraph constraints

Abstract

Let $k,l$ be two positive integers. An $S_{k,l}$ is a graph obtained from disjoint $K_{1,k}$ and $K_{1,l}$ by adding an edge between the $k$-degree vertex in $K_{1,k}$ and the $l$-degree vertex in $K_{1,l}$. An {\em $S_{k,l}$-free} graph is a graph containing no induced subgraph isomorphic to $S_{k,l}$. In this note, we show that, for any positive integers $k,l$ with $2\leqslant k\leqslant l$, there exists a constant $c=c(k,l)$ such that every connected balanced $S_{k,l}$-free bipartite graph with minimum degree at least $c$ contains a connected $k$-factor.