The existence of $m$-tree-connected $(g,f+f'-m)$-factors using $(g,f)$-factors and $m$-tree-connected $(m,f')$-factors

TL;DR

This paper extends existing results on graph factors by demonstrating that under certain conditions, a graph containing specific factors also contains an $m$-tree-connected factor with combined properties, broadening the applicability of factor existence theorems.

Contribution

The paper generalizes previous factor existence results to $m$-tree-connected factors, allowing for nonnegative $g$ and establishing new conditions for their existence.

Findings

Proves the existence of $m$-tree-connected $(g,f+f'-m)$-factors under specified conditions.

Extends prior work by relaxing the requirement on $g$, allowing it to be nonnegative.

Provides a new framework for constructing $m$-tree-connected factors in graphs.

Abstract

Let $G$ be a graph and let $g$, $f$, and $f'$ be three positive integer-valued functions on $V(G)$ with $g\le f$. Tokuda, Xu, and Wang (2003) showed that if $G$ contains a $(g,f)$-factor and a spanning $f'$-tree, then $G$ also contains a connected $(g,f+f'-1)$-factor. In this note, we develop their result to a tree-connected version by proving that if $G$ contains a $(g,f)$-factor and an $m$-tree-connected $(m,f')$-factor, then $G$ also contains an $m$-tree-connected $(g,f+f'-m)$-factor, provided that $f\ge m$. In addition, we show that $g$ allows to be nonnegative.