Degree Factors with Red-Blue Coloring of Regular Graphs

Abstract

Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following:
 Let $a$, $b$ and $k$ be integers with $1\leq a\leq k\leq b\leq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $\deg_{F}(x)\in \{a,b\}$ for every red vertex $x$ and $\deg_{F}(y)\in \{k,k+1\}$ for every blue vertex $y$.