Defective coloring of blowups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910901728182272 |
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| author | Norin, Sergey Steiner, Raphael |
| author_facet | Norin, Sergey Steiner, Raphael |
| contents | Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $χ^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $χ(G)=χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $χ(G)\ge \frac{30}{29}χ^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $χ^d(G\boxtimes K_{d+1})\le χ(G)\le 2χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Defective coloring of blowups Norin, Sergey Steiner, Raphael Combinatorics 05C07, 05C15, 05C76 Given a graph $G$ and an integer $d\ge 0$, its $d$-defective chromatic number $χ^d(G)$ is the smallest size of a partition of the vertices into parts inducing subgraphs with maximum degree at most $d$. Guo, Kang and Zwaneveld recently studied the relationship between the $d$-defective chromatic number of the $(d+1)$-fold (clique) blowup $G\boxtimes K_{d+1}$ of a graph $G$ and its ordinary chromatic number, and conjectured that $χ(G)=χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. In this note we disprove this conjecture by constructing graphs $G$ of arbitrarily large chromatic number such that $χ(G)\ge \frac{30}{29}χ^d(G\boxtimes K_{d+1})$ for infinitely many $d$. On the positive side, we show that the conjecture holds with a constant factor correction, namely $χ^d(G\boxtimes K_{d+1})\le χ(G)\le 2χ^d(G\boxtimes K_{d+1})$ for every graph $G$ and $d\ge 0$. |
| title | Defective coloring of blowups |
| topic | Combinatorics 05C07, 05C15, 05C76 |
| url | https://arxiv.org/abs/2504.01548 |