关键词: $b$-染色, $b$-染色数, $b$-连续, $m$-度, Corona图, 循环染色

Abstract: Let $\{V_{1}, V_{2},\cdots, V_{k}\}$ be a proper vertex coloring of a graph $G=(V,E)$, which is called a $b$-coloring of $G$, if for all $i, j: 1\leq i\neq j\leq k$, exists $u\in V_{i}, v\in V_{j}$, satisfying $uv\in E$. The maximum positive integer $k$ for a $b$-coloring $\{V_{1}, V_{2},\cdots, V_{k}\}$ on a graph $G$ is called the $b$-chromatic number, denoted by $b(G)$. A graph G is called $b$-continuity if for all $k:\chi(G) \leq k \leq b(G)$, there exists a $(k)b$-coloring on graph $G$. According to the structural characteristics of the Corona graphs, the  cyclic coloring schemes are constructed. Through the cyclic coloring on two kinds of vertices of Corona graphs, the $b$-chromatic number of several Corona graphs equalling to its $m$-degree is obtained, and all these Corona graphs are $b$-continuous.

Key words: $b$-coloring, $b$-chromatic number, $b$-continuity, $m$-degree, Corona graph, cyclic coloring scheme