Distribution of Vertices Required a High-Degree Condition on Partitions of Graphs Under Degree Constraints

Abstract

Let $G$ be a graph, and let $f_1,f_2:V\left(G\right)\to \left\{0,1,2,\dots \right\}$ be functions. Let $T_1\left(G\right)$ be the union of edges shared by two cycles of order at most four, and let $T_0\left(G\right)=V\left(G\right)\backslash T_1\left(G\right)$. In this paper, we prove that if for $u\in T_h\left(G\right)$ with $h\in \left\{0,1\right\}$, $d_G\left(u\right)\ge f_1\left(u\right)+f_2\left(u\right)-1+2h$ and $\min \left\{f_1\left(u\right),f_2\left(u\right)\right\}\ge 2-2h$, then $G$ can be partitioned into two subgraphs $G_1$ and $G_2$ such that $d_{G_i}\left(u\right)\ge f_i\left(u\right)$ for each $u\in V\left(G_i\right)$. The result is a generalization of some known results and gives a distribution of vertices required by a high-degree condition on partitions of graphs under degree constraints.