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- W3027992394 abstract "Given a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,ldots)$ of positive integers, the mapping $f:V(G) rightarrow {1,ldots,k}$ is an $S$-packing $k$-coloring of $G$ if for any distinct vertices $u,vin V(G)$ with $f(u)=f(v)=i$ the distance between $u$ and $v$ in $G$ is greater than $a_i$. The smallest $k$ such that $G$ has an $S$-packing $k$-coloring is the $S$-packing chromatic number, $chi_S(G)$, of $G$. In this paper, we consider the distance graphs $G(mathbb{Z},{2,t})$, where $t>1$ is an odd integer, which has $mathbb{Z}$ as its vertex set, and $i,jinmathbb{Z}$ are adjacent if $|i-j|in{2,t}$. We determine the $S$-packing chromatic numbers of the graphs $G(mathbb{Z},{2,t})$, where $S$ is any sequence with $a_iin{1,2}$ for all $i$. In addition, we give lower and upper bounds for the $d$-distance chromatic numbers of the distance graphs $G(mathbb{Z},{2,t})$, which in the cases $dge t-3$ give the exact values. Implications for the corresponding $S$-packing chromatic numbers of the circulant graphs are also discussed." @default.
- W3027992394 created "2020-05-29" @default.
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- W3027992394 date "2020-05-21" @default.
- W3027992394 modified "2023-09-27" @default.
- W3027992394 title "$S$-packing colorings of distance graphs $G(mathbb{Z},{2,t})$" @default.
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