Further Results and Questions on $S$-Packing Coloring of Subcubic Graphs

TL;DR

This work advances S-packing coloring within subcubic graphs by introducing saturation-based graph classes and proving new positive results. It provides a novel proof technique for $1$-saturated graphs, leveraging a maximum weighted independent set and a detailed structural decomposition of the complement graph into maximal paths to obtain a $(1,1,3,3)$-packing coloring, and also demonstrates $(1,2^4)$- and $(1,2^5)$-packing colorings for related saturated classes. These results extend the landscape of packing colorings for subcubic graphs and connect to broader questions about the packing chromatic number of subdivisions. The conclusions outline open problems, notably improving colorings to $(1,1,3,4)$ and optimizing the independent-set weighting to sharpen the method’s reach.

Abstract

For non-decreasing sequence of integers $S=(a_1,a_2, \dots, a_k)$, an $S$-packing coloring of $G$ is a partition of $V(G)$ into $k$ subsets $V_1,V_2,\dots,V_k$ such that the distance between any two distinct vertices $x,y \in V_i$ is at least $a_{i}+1$, $1\leq i\leq k$. We consider the $S$-packing coloring problem on subclasses of subcubic graphs: For $0\le i\le 3$, a subcubic graph $G$ is said to be $i$-saturated if every vertex of degree 3 is adjacent to at most $i$ vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and $G$ is said to be $(3,i)$-saturated if every heavy vertex is adjacent to at most $i$ heavy vertices. We prove that every 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable. We also prove that every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable.

Figures (4)

• Figure 1: A non $(1,1,4,4)$-packing colorable $1$-saturated graph (on the left) and a non $(1,1,3,3)$-packing colorable $(3,2)$-saturated subcubic graph (on the right).
• Figure 2: The different types of maximal paths in $G[\overline{S}]$.
• Figure 3: A $1$-saturated non $(1,2^3)$-packing coloring subcubic graph.
• Figure 4: Two configurations of 6 non-heavy vertices $x_i$ at pairwise distance at most two in a $(3,0)$-saturated subcubic graph, along with a $(1,2^4)$-packing coloring for each.

Theorems & Definitions (11)

• Theorem 1
• Claim 1.1
• Claim 1.2
• Claim 1.3
• Claim 1.4
• Claim 1.5
• Lemma 1
• Lemma 2
• Theorem 2
• Theorem 3