On the star b-chromatic number of a graph

TL;DR

The paper introduces the star b-chromatic number $S_b(G)$, a dual concept to star colorings defined via a star-recoloring partial order, and investigates its fundamental properties and bounds. It develops the notion of star degree $d_G^s(v)$ and derives a girth-based, nearly complete formula when $g(v)\ge7$, enabling upper bounds and extremal constructions. A key upper bound $S_b(G)\le m_s(G)$ is established, with $m_s(G)$ further bounded by $ abla(G)^2+1$, and explicit extremal tree families are constructed where equality is attained. The authors compute exact $S_b(G)$ values for several graph families (paths, cycles, joins) and demonstrate that $S_b(G)$ can be arbitrarily larger than classical invariants such as $ abla(G)$ and $S(G)$, via various graph constructions. They conclude with open questions on computational complexity, especially for trees and general graphs, and invite further study of $S_b(T)$ for trees and related classes, highlighting the significance of $S_b(G)$ as a robust measure tied to star colorings and recoloring dynamics.

Abstract

A star coloring of a graph $G$ is a proper coloring where vertices of every two color classes induce a forest of stars. A strict partial order is defined on the set of all star colorings of $G$. We introduce the star b-chromatic number $S_b(G)$, analogous to the b-chromatic number, as the maximum number of colors in a minimum element of the mentioned order. We present several combinatorial properties of $S_b(G)$, compute the exact value for $S_b(G)$ for several known families and compare $S_b(G)$ with several invariants naturally connected to $S_b(G)$.

Figures (6)

• Figure 1: Two star colorings of a graph. In the first coloring the black vertices have all the other colors blocked, meaning that every color class has at least one star b-vertex. For the vertices that are not strong star b-vertices, the available colors are indicated. The second coloring differs from the first only in $v$, which is indicated in brackets. This also changes the available colors for vertex $u$, allowing us to apply a star recoloring step for color $7$.
• Figure 2: Blocking paths $0123$, $0122$ and $1012$, shown from left to right.
• Figure 3: Graph $G$ with $d_G^s(v)=17$ which corresponds to a partition $A_1(v)=\{a_1^1,a_6^1,a_9^1 \}$, $A_2(v)=\{a_2^2,a_3^2,a_4^2,a_7^2\}$, $A_3(v)=\{a_5^3,a_8^3 \}$ where $X(v)=\{b_1,b_2,b_5,b_7,b_8\}$ and $Y(v)=\{b_3,b_4,b_6,b_9,b_{10}\}$.
• Figure 4: Vertex $v$ with $d_G^s(v)=11$ where colors $9$ and $10$ are blocked by $0122$ blocking paths.
• Figure 5: Graph $T_3$ and its star b-coloring with $10$ colors.

Theorems & Definitions (18)

• Proposition 3.1
• Proposition 3.2
• Definition 3.3
• Theorem 3.4
• Corollary 3.5
• Theorem 4.1
• Theorem 4.2
• Theorem 5.1
• Proposition 5.2
• Corollary 5.3