On $S$-packing Coloring of Bounded Degree Graphs

TL;DR

This work investigates $S$-packing coloring on graphs with bounded maximum degree, extending results from subcubic graphs to general $k$ by introducing $0$-, $t$-, and $(k-1)$-saturated graph classes. It develops a cohesive framework based on maximal independent sets, heavy/father/sibling structure, and Brooks' theorem applied to graph powers like $G^2$ to construct explicit colorings, achieving $(1^{k-1},2^k)$-packing colorability for general $k$ and refined results for saturated subclasses. The authors present comprehensive proofs for each saturation level, culminating in a general $k$-degree result and a discussion of sharpness, with conjectures and open questions that chart directions for future work. The results have implications for understanding how distance constraints interact with degree bounds to govern feasible packings in large classes of graphs.

Abstract

Given a sequence $S=(s_1,s_2,\ldots,s_p)$, $p\geq 2$, of non-decreasing integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $p$ disjoint sets $V_1,\ldots, V_p$ such that any two distinct vertices of $V_i$ are at a distance greater than $s_i$, $1\le i\le p$. In this paper, we study the $S$-packing coloring problem on graphs of bounded maximum degree and for sequences mainly containing 1's and 2's ($i^r$ in a sequence means $i$ is repeated $r$ times). Generalizing existing results for subcubic graphs, we prove a series of results on graphs of maximum degree $k$: We show that graphs of maximum degree $k$ are $(1^{k-1},2^k)$-packing colorable. Moreover, we refine this result for restricted subclasses: A graph of maximum degree $k$ is said to be $t$-saturated, $0\le t\le k$, if every vertex of degree $k$ is adjacent to at most $t$ vertices of degree $k$. We prove that any graph of maximum degree $k\ge 3$ is $(1^{k-1}, 3)$-packing colorable if it is 0-saturated, $(1^{k-1}, 2)$-packing colorable if it is $t$-saturated, $1\leq t\leq k-2$; and $(1^{k-1},2^{k-1})$-packing colorable if it is $(k-1)$-saturated. We also propose some conjectures and questions.

Figures (2)

• Figure 1: The different types of maximal paths in $G[\overline{S}]$.
• Figure 2: The graph $G_{5,3}$ is a $3$-saturated $5$-degree graph that is not $(1^4,3^3)$-packing colorable.

Theorems & Definitions (26)

• Theorem 2.1
• Theorem 3.1
• Remark 3.1
• Claim 3.1.1
• Remark 3.2
• Claim 3.1.2
• Theorem 4.1
• Claim 4.1.1
• Remark 4.1
• Remark 4.2