Every connected subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable

Abstract

For a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, a packing $S$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, V_2, \ldots, V_k$ such that each $V_i$ has pairwise distance at least $s_i+1$. The packing chromatic number (PCN) of a graph $G$ is the minimum $k$ such that $G$ has a packing $(1,2, \ldots, k)$-coloring. The $1$-subdivision of $G$ is obtained by replacing each edge of $G$ with a path of two edges. In 2016, Gastineau and Togni asked an open question whether the $1$-subdivision of every subcubic graph has PCN at most $5$, and later Bre\v sar, Klav\v zar, Rall, and Wash conjectured it is true. Balogh, Kostochka, and Liu proved the first upper bound of $8$, and it was later improved to $6$ by Liu, Zhang, and Zhang. In this paper, we prove that every connected subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable. Our result implies a solution to the conjecture of Bre\v sar, Klav\v zar, Rall, and Wash, and answers the question of Gastineau and Togni in the affirmative. Furthermore, our result answers an open question of Kostochka and Liu and solves a conjecture of Liu, Zhang, and Zhang.

Figures (4)

• Figure 1: The structures in $G$ corresponding to the cycle components of order 3 in $H$.
• Figure 2: Configurations $F_1$ and $F_2$.
• Figure 3: Form the Petersen graph.
• Figure 4: The interfaces of path component of $H$ and the structure used in Lemma \ref{['admissthird']}.

Theorems & Definitions (25)

• Conjecture 2: Brešar, Klavžar, Rall, and Wash BKRW1
• Conjecture 4: Liu, Zhang, and Zhang LZZ1
• Theorem 5
• Lemma 6
• Lemma 8
• proof
• Remark 9
• Lemma 10
• proof
• Lemma 11