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arXiv · 2505.17369

S-packing chromatic critical graphs

Gülnaz Boruzanlı Ekinci·Csilla Bujtás·Didem Gözüpek·Sandi Klavžar
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Abstract

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $χ_S(G)$. In this paper, $χ_S$-critical graphs are introduced as the graphs $G$ such that $χ_S(H) < χ_S(G)$ for each proper subgraph $H$ of $G$. Several families of $χ_S$-critical graphs are constructed, and $2$- and $3$-colorable $χ_S$-critical graphs are presented for all packing sequences $S$, while $4$-colorable $χ_S$-critical graphs are found for most of $S$. Cycles which are $χ_S$-critical are characterized under different conditions. It is proved that for any graph $G$ and any edge $e \in E(G)$, the inequality $χ_S(G - e) \ge χ_S(G)/2$ holds. Moreover, in several important cases, this bound can be improved to $χ_S(G - e) \ge (χ_S(G)+1)/2$. The sharpness of the bounds is also discussed. Along the way an earlier result on $χ_S$-vertex-critical graphs is supplemented.

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Gülnaz Boruzanlı Ekinci, Csilla Bujtás, Didem Gözüpek, Sandi Klavžar. 2025-05-23. S-packing chromatic critical graphs. https://arxiv.org/abs/2505.17369

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