Nearly optimal coloring of some C4-free graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929496302551040 |
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| author | Chen, Ran Xu, Baogang |
| author_facet | Chen, Ran Xu, Baogang |
| contents | A class ${\cal G}$ of graphs is $χ$-{\em polydet} if ${\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(ω(G))$-coloring of $G\in {\cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {\em bull} consists of a triangle with two disjoint pendant edges, a {\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_3, C_4, F)$-free graphs without clique cutsets and universal cliques, and present a close relation between $(C_4, F, H)$-free graphs and the Petersen graph. As a consequence, we show that the classes of $(C_4, F, H)$-free graphs are $χ$-polydet with nearly optimal linear binding functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nearly optimal coloring of some C4-free graphs Chen, Ran Xu, Baogang Combinatorics A class ${\cal G}$ of graphs is $χ$-{\em polydet} if ${\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(ω(G))$-coloring of $G\in {\cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {\em bull} consists of a triangle with two disjoint pendant edges, a {\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_3, C_4, F)$-free graphs without clique cutsets and universal cliques, and present a close relation between $(C_4, F, H)$-free graphs and the Petersen graph. As a consequence, we show that the classes of $(C_4, F, H)$-free graphs are $χ$-polydet with nearly optimal linear binding functions. |
| title | Nearly optimal coloring of some C4-free graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.06944 |