S-packing chromatic critical graphs

TL;DR

This work generalizes packing coloring through $S$-packing colorings and introduces $χ_S$-critical graphs, exploring their structure and extremal behavior. It provides foundational properties (connectivity, vertex-criticality) and complete classifications for $2$- and $3$-$χ_S$-critical graphs, plus substantial results for $4$-$χ_S$-critical graphs across important families of packing sequences, including a precise cycle analysis. A central result is a universal lower bound $χ_S(G-e)\ge χ_S(G)/2$ upon edge removal, with sharper bounds in many cases and multiple sharp examples demonstrating optimality. The paper also bridges to $χ_S$-vertex-critical graphs and outlines open problems, especially in remaining S-classes, motivating further study of $S$-packing critical graphs and cycle behavior.

Abstract

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $χ_S(G)$. In this paper, $χ_S$-critical graphs are introduced as the graphs $G$ such that $χ_S(H) < χ_S(G)$ for each proper subgraph $H$ of $G$. Several families of $χ_S$-critical graphs are constructed, and $2$- and $3$-colorable $χ_S$-critical graphs are presented for all packing sequences $S$, while $4$-colorable $χ_S$-critical graphs are found for most of $S$. Cycles which are $χ_S$-critical are characterized under different conditions. It is proved that for any graph $G$ and any edge $e \in E(G)$, the inequality $χ_S(G - e) \ge χ_S(G)/2$ holds. Moreover, in several important cases, this bound can be improved to $χ_S(G - e) \ge (χ_S(G)+1)/2$. The sharpness of the bounds is also discussed. Along the way an earlier result on $χ_S$-vertex-critical graphs is supplemented.

Figures (1)

• Figure 1: The graphs $G_1, \ldots, G_8$

Theorems & Definitions (16)

• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Corollary 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Theorem 3.8
• Theorem 3.9
• Proposition 4.1