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- W2985250243 abstract "For a sequence of non-decreasing positive integers $S = (s_1, ldots, s_k)$, a packing $S$-coloring is a partition of $V(G)$ into sets $V_1, ldots, V_k$ such that for each $1leq i leq k$ the distance between any two distinct $x,yin V_i$ is at least $s_i+1$. The smallest $k$ such that $G$ has a packing $(1,2, ldots, k)$-coloring is called the packing chromatic number of $G$ and is denoted by $chi_p(G)$. For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge. The question whether $chi_p(D(G)) le 5$ for all subcubic graphs was first asked by Gastineau and Togni and later conjectured by Bresar, Klavzar, Rall and Wash. Gastineau and Togni observed that if one can prove every subcubic graph except the Petersen graph is packing $(1,1,2,2)$-colorable then the conjecture holds. The maximum average degree, mad($G$), is defined to be $max{frac{2|E(H)|}{|V(H)|}: H subset G}$. In this paper, we prove that subcubic graphs with $mad(G)<frac{30}{11}$ are packing $(1,1,2,2)$-colorable. As a corollary, the conjecture of Bresar et al holds for every subcubic graph $G$ with $mad(G)<frac{30}{11}$." @default.
- W2985250243 created "2019-11-22" @default.
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- W2985250243 date "2019-11-09" @default.
- W2985250243 modified "2023-09-27" @default.
- W2985250243 title "Packing $(1,1,2,2)$-coloring of some subcubic graphs" @default.
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- W2985250243 doi "https://doi.org/10.48550/arxiv.1911.03824" @default.
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