Brooks-type theorems for relaxations of square colorings
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910412542312448 |
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| author | Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram |
| author_facet | Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram |
| contents | The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer $h$, the proper $h$-conflict-free chromatic number of a graph $G$, denoted $χ_{pcf}^h(G)$, is the minimum $k$ such that $G$ has a proper $k$-coloring where every vertex $v$ has $\min\{deg_G(v),h\}$ colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if $G$ is a graph with $Δ(G)\ge 3$, then $χ_{pcf}^1(G)\leq Δ(G)+1$. The best known result regarding the conjecture is $χ_{pcf}^1(G)\leq 2Δ(G)+1$, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all $h$, and also enlarge the class of graphs for which the conjecture is known to be true.
Our main result is the following: for a graph $G$, if $Δ(G) \ge h+2$, then $χ_{pcf}^h(G)\le (h+1)Δ(G)-1$; this is tight up to the additive term as we explicitly construct infinitely many graphs $G$ with $χ_{pcf}^h(G)=(h+1)(Δ(G)-1)$. We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for $h$-dynamic coloring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06125 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Brooks-type theorems for relaxations of square colorings Cho, Eun-Kyung Choi, Ilkyoo Kwon, Hyemin Park, Boram Combinatorics The following relaxation of proper coloring the square of a graph was recently introduced: for a positive integer $h$, the proper $h$-conflict-free chromatic number of a graph $G$, denoted $χ_{pcf}^h(G)$, is the minimum $k$ such that $G$ has a proper $k$-coloring where every vertex $v$ has $\min\{deg_G(v),h\}$ colors appearing exactly once on its neighborhood. Caro, Petruševski, and Škrekovski put forth a Brooks-type conjecture: if $G$ is a graph with $Δ(G)\ge 3$, then $χ_{pcf}^1(G)\leq Δ(G)+1$. The best known result regarding the conjecture is $χ_{pcf}^1(G)\leq 2Δ(G)+1$, which is implied by a result of Pach and Tardos. We improve upon the aforementioned result for all $h$, and also enlarge the class of graphs for which the conjecture is known to be true. Our main result is the following: for a graph $G$, if $Δ(G) \ge h+2$, then $χ_{pcf}^h(G)\le (h+1)Δ(G)-1$; this is tight up to the additive term as we explicitly construct infinitely many graphs $G$ with $χ_{pcf}^h(G)=(h+1)(Δ(G)-1)$. We also show that the conjecture is true for chordal graphs, and obtain partial results for quasi-line graphs and claw-free graphs. Our main result also improves upon a Brooks-type result for $h$-dynamic coloring. |
| title | Brooks-type theorems for relaxations of square colorings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2302.06125 |