关于图的字典序积的b-连续性
On the b-continuity of the lexicographic product of graphs
摘要：
一个图的顶点的b-染色是一个正常染色，其中每个颜色类包含一个顶点，该顶点与其他颜色类相邻。$G$的b-色数是使$G$具有带$χ_b(G)$颜色的b-染色的最大整数$χ_b(G)$。一个图$G$是b-连续的，如果对区间$[χ(G),χ_b(G)]$中的每一个整数$k$，$G$都有一个用$k$颜色的b-染色.众所周知，并非所有的图都是b-连续的。在这里，我们研究是否字典积$G[H]$的b-连续图$G$和$H$也是b-连续的。利用同态，我们给出了$χ_b(G[H])$的一个新的下界，即$χ_b(G[K_t])$，其中$t=χ_b(H)$，并证明了：如果$G[K_\ell]$对每个正整数$\ell$都是b-连续的，则$G[H]$对区间$[χ(G[H]),χ_b(G[K_t])]$中的每个$k$都允许用$k$颜色进行b-染色.证明了当$G$是$P_4$ -稀疏图时，对任意正整数$\ell$，$G[K_\ell]$是b-连续的;当$G$是弦图时，给出了$G[K_\ell]$的b-谱的进一步结果.
英文摘要：
A b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to each other color class. The b-chromatic number of $G$ is the maximum integer $χ_b(G)$ for which $G$ has a b-coloring with $χ_b(G)$ colors. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[χ(G),χ_b(G)]$. It is known that not all graphs are b-continuous. Here, we investigate whether the lexicographic product $G[H]$ of b-continuous graphs $G$ and $H$ is also b-continuous. Using homomorphisms, we provide a new lower bound for $χ_b(G[H])$, namely $χ_b(G[K_t])$, where $t=χ_b(H)$, and prove that if $G[K_\ell]$ is b-continuous for every positive integer $\ell$, then $G[H]$ admits a b-coloring with $k$ colors, for every $k$ in the interval $[χ(G[H]),χ_b(G[K_t])]$. We also prove that $G[K_\ell]$ is b-continuous, for every positive integer $\ell$, whenever $G$ is a $P_4$-sparse graph, and we give further results on the b-spectrum of $G[K_\ell]$, when $G$ is chordal.
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