On the b-continuity of the lexicographic product of graphs

TL;DR

This paper investigates the b-continuity property of the lexicographic product of graphs, establishing conditions under which the product maintains b-continuity and providing bounds for the b-chromatic number.

Contribution

It introduces a new lower bound for the b-chromatic number of lexicographic products and proves b-continuity for certain classes of graphs, such as P4-sparse and chordal graphs.

Findings

G[G] is b-continuous for P4-sparse graphs G.

A new lower bound for the b-chromatic number of G[H] is established.

Conditions under which G[H] maintains b-continuity are identified.

Abstract

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 $\chi_b(G)$ for which $G$ has a b-coloring with $\chi_b(G)$ colors. A graph $G$ is b-continuous if $G$ has a b-coloring with $k$ colors, for every integer $k$ in the interval $[\chi(G),\chi_b(G)]$. 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 $\chi_b(G[H])$, namely $\chi_b(G[K_t])$, where $t=\chi_b(H)$, 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 $[\chi(G[H]),\chi_b(G[K_t])]$. 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.