Notes on complexity of packing coloring

Minki Kim; Bernard Lidický; Tomáš Masařík; Florian Pfender

doi:10.1016/j.ipl.2018.04.012

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arXiv · 1712.08373

Notes on complexity of packing coloring

Minki Kim·Bernard Lidický·Tomáš Masařík·Florian Pfender

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Abstract

A packing $k$-coloring for some integer $k$ of a graph $G=(V,E)$ is a mapping $\varphi:V\to\{1,\ldots,k\}$ such that any two vertices $u, v$ of color $\varphi(u)=\varphi(v)$ are in distance at least $\varphi(u)+1$. This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of $G$ is the smallest $k$ such that there exists a packing $k$-coloring of $G$. Fiala and Golovach showed that determining the packing chromatic number for chordal graphs is \NP-complete for diameter exactly 5. While the problem is easy to solve for diameter 2, we show \NP-completeness for any diameter at least 3. Our reduction also shows that the packing chromatic number is hard to approximate within $n^{{1/2}-\varepsilon}$ for any $\varepsilon > 0$. In addition, we design an \FPT algorithm for interval graphs of bounded diameter. This leads us to exploring the problem of finding a partial coloring that maximizes the number of colored vertices.

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Minki Kim, Bernard Lidický, Tomáš Masařík, Florian Pfender. 2017-12-22. Notes on complexity of packing coloring. https://doi.org/10.1016/j.ipl.2018.04.012

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