The existence of $m$-tree-connected $(g,f+f^{\prime }-m)$-factors using $(g,f)$-factors and $m$-tree-connected $(m,f^{\prime })$-factors

Abstract

Let $G$ be a graph and let $g$, $f$, and $f^{\prime }$ 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 $f^{\prime }$-tree, then $G$ also contains a connected $(g,f+f^{\prime }-1)$-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 $(m,f^{\prime })$-factor, then $G$ also contains an $m$-tree-connected $(g,f+f^{\prime }-m)$-factor, provided that $f\ge m$. In addition, we show that $g$ allows to be nonnegative.

This paper is an improved version of a removed part of the paper arXiv:1702.06203