On the b-chromatic number of star graph operators

TL;DR

The paper analyzes the $b$-chromatic number for star-graph operators, focusing on the Cartesian product of two stars, its graph powers, and derived line/total graphs. It provides constructive colorings and tight bounds, proving exact values such as $\varphi(S_n\square S_m)=m+2$ and detailing the behavior of $\varphi$ for power graphs across several regimes of $n,m$ and $k$. The results connect to rook graphs and $K_n\square K_m$, offering a comprehensive view of $b$-colorings on these structured graph families and presenting several open questions for further study. This advances the theory of $b$-colorings in graph products and graph operators with potential implications for network design and combinatorial optimization.

Abstract

A $b$-coloring is a proper coloring such that for each color class, there exists at least one vertex that is adjacent to at least one vertex in every other color class. The $b$-chromatic number of a graph $G$ is the maximum number $k$ such that $G$ admits a $b$-coloring with $k$ colors. This paper focuses on the $b$-chromatic number of the power graph of the Cartesian product of star graphs. In addition, we also study the total graph and the line graph of the Cartesian product of star graphs. Our main result generalizes the result shown in \cite{qn} on the b-chromatic number of the Cartesian product of two stars. We find exact values for the b-chromatic number of particular Cartesian products of complete graphs and explore the bounds of the generalized Cartesian product of complete graphs.

Figures (12)

• Figure 1: Proper coloring of $P_4$.
• Figure 2: $b$-coloring of $G$ such that $\varphi(G)=5$
• Figure 3: The Cartesian Product of $P_3$ and $P_3$.
• Figure 4: The second power of $P_4$.
• Figure 5: $S_4\square S_3$

Theorems & Definitions (50)

• Example 1.1
• Definition 1
• Example 1.2
• Theorem 1.1: Theorem 4.2
• Theorem 1.2: Theorem 3.1
• Theorem 1.3: Theorem 3.2
• Theorem 1.4: Theorem 3.3
• Definition 2
• Proposition 2.1
• Proposition 2.2