Notes on complexity of packing coloring
Minki Kim, Bernard Lidick\'y, Tom\'a\v{s} Masa\v{r}\'ik, Florian, Pfender

TL;DR
This paper investigates the computational complexity of packing coloring in graphs, proving NP-completeness for various diameters, and introduces an FPT algorithm for interval graphs with bounded diameter, advancing understanding of the problem's difficulty.
Contribution
The paper establishes NP-completeness of packing chromatic number for graphs with diameter at least 3 and develops an FPT algorithm for interval graphs of bounded diameter.
Findings
NP-completeness for diameter ≥ 3
Hard to approximate within n^{1/2 - ε}
FPT algorithm for interval graphs with bounded diameter
Abstract
A packing -coloring for some integer of a graph is a mapping such that any two vertices of color are in distance at least . This concept is motivated by frequency assignment problems. The \emph{packing chromatic number} of is the smallest such that there exists a packing -coloring of . 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 for any . In addition, we design an \FPT algorithm for interval graphs of bounded diameter. This leads us to exploring the problem of…
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