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Impact of local girth on the S-packing coloring of k-saturated subcubic graphs

Ayman El Zein, Maidoun Mortada

Abstract

For a non-decreasing sequence $S=(s_1,s_2,\dots,s_k)$, an $S$-packing coloring of a graph $G$ is a vertex coloring using the colors $s_1,s_2,\dots,s_k$ such that any two vertices assigned the same color $s_i$ are at distance greater than $s_i$. A subcubic graph is said to be $k$-saturated, for $0\le k\le3$, if every vertex of degree 3 is adjacent to at most $k$ vertices of degree~3. The \emph{local girth} of a vertex is the length of the smallest cycle containing it. Brešar, Kuenzel, and Rall [\textit{Discrete Math.} 348(8) (2025),~114477] proved that every claw-free cubic graph is $(1,1,2,2)$-packing colorable, confirming the conjecture for this family. Equivalently, a claw-free cubic graph is one in which each $3$-vertex has local girth~3. Motivated by this observation and by recent progress on $S$-packing colorings of $k$-saturated subcubic graphs, we study the influence of local girth on their $S$-packing colorability. We establish a series of results describing how the parameters of saturation and local girth jointly determine the admissible $S$-packing sequences. Sharpness is verified through explicit constructions, and several open problems are posed to delineate the remaining cases.

Impact of local girth on the S-packing coloring of k-saturated subcubic graphs

Abstract

For a non-decreasing sequence , an -packing coloring of a graph is a vertex coloring using the colors such that any two vertices assigned the same color are at distance greater than . A subcubic graph is said to be -saturated, for , if every vertex of degree 3 is adjacent to at most vertices of degree~3. The \emph{local girth} of a vertex is the length of the smallest cycle containing it. Brešar, Kuenzel, and Rall [\textit{Discrete Math.} 348(8) (2025),~114477] proved that every claw-free cubic graph is -packing colorable, confirming the conjecture for this family. Equivalently, a claw-free cubic graph is one in which each -vertex has local girth~3. Motivated by this observation and by recent progress on -packing colorings of -saturated subcubic graphs, we study the influence of local girth on their -packing colorability. We establish a series of results describing how the parameters of saturation and local girth jointly determine the admissible -packing sequences. Sharpness is verified through explicit constructions, and several open problems are posed to delineate the remaining cases.

Paper Structure

This paper contains 5 sections, 24 theorems, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. $0$-Saturated and $1$-Saturated
  3. $2$-Saturated
  4. $3$-Saturated
  5. Sharpness of the Results and Open Problems

Key Result

Lemma 1

Let $G$ be a connected graph such that $\Delta(G)\leq 2$. Then,

Figures (5)

  • Figure 1: The red paths are either of length one or two. The vertex $y$ (resp., $z$, $w$) is one of the vertices $y_i$ (resp., $z_i$, $w_i$).
  • Figure 2: The graph $\mathcal{G}_1$: a $0$-saturated subcubic graph such that $g_3(\mathcal{G}_1)=4$ that is neither $(1,2,2)$-packing colorable nor $(2,2,2,2,2)$-packing colorable. The graph $\mathcal{G}_2$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_2)=3$ that is not $(1,2,2)$-packing colorable. The graph $\mathcal{G}_3$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_3)=3$.
  • Figure 3: The graph $\mathcal{G}_4$: a $0$-saturated subcubic graph such that $g_3(\mathcal{G}_4)=4$ that is not $(2,2,2,2)$-packing colorable. The graph $\mathcal{G}_5$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_5)=3$ that is not $(1,1,3)$-packing colorable. The graph $\mathcal{G}_6$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_6)=3$ that is not $(2,2,2,2)$-packing colorable.
  • Figure 4: The graph $\mathcal{G}_7$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_7)=4$ that is not $(1,2,2)$-packing colorable. The graph $\mathcal{G}_8$: a $1$-saturated subcubic graph such that $g_3(\mathcal{G}_8)=5$ that is not $(1,2,2,2)$-packing colorable. The graph $\mathcal{G}_9$: a $2$-saturated subcubic graph such that $g_3(\mathcal{G}_9)=4$ that is neither $(2,2,2,2,2)$-packing colorable nor $(1,2,2,2)$-packing colorable.
  • Figure 5: The graph $\mathcal{G}_{10}$: a $2$-saturated subcubic graph such that $g_3(\mathcal{G}_{10})=3$ that is neither $(1,1,3,3)$-packing colorable nor $(1,2,2,3)$-packing colorable. The graph $\mathcal{G}_{11}$: a $(3,3)$-saturated subcubic graph such that $g_3(\mathcal{G}_{11})=3$ that is not $(1,1,3,3,3)$-packing colorable.

Theorems & Definitions (65)

  • Lemma 1
  • Theorem 1
  • Claim 1.1
  • Claim 1.2
  • Claim 1.3
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Lemma 2
  • Claim 2.1
  • ...and 55 more