Degree Factors with Red-Blue Coloring of Regular Graphs

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 S a S k S b S k + a + 1, and let r be a sufficiently large integer compared to a , b and k. Let G be an r -regular graph. For every red -blue vertex coloring of G in which no two red vertices are adjacent, there exists a factor F of G such that degF(x) is an element of {a, b} for every red vertex x and degF(y) is an element of {k, k + 1} for every blue vertex y.