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

TL;DR

This paper investigates $S$-packing colorings in subcubic graphs, establishing new coloring bounds for specific subclasses characterized by saturation and heaviness conditions.

Contribution

It introduces new results on $S$-packing colorability for subclasses of subcubic graphs based on saturation and heaviness constraints.

Findings

1-saturated graphs are $(1,1,3,3)$-packing colorable

1-saturated graphs are $(1,2,2,2,2)$-packing colorable

$(3,0)$-saturated graphs are $(1,2,2,2,2,2)$-packing colorable

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.