Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Ekinci, Gülnaz Boruzanlı, Bujtás, Csilla, Gözüpek, Didem, Klavžar, Sandi
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979328469
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