Coloring graphs with forbidden almost bipartite subgraphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910935817388032 |
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| author | Anderson, James Bernshteyn, Anton Dhawan, Abhishek |
| author_facet | Anderson, James Bernshteyn, Anton Dhawan, Abhishek |
| contents | Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $χ(G) \leq (c(F) + o(1)) Δ/ \logΔ$ whenever $G$ is an $F$-free graph of maximum degree $Δ$. The largest class of connected graphs $F$ for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph $K_{1,t,t}$ for some $t \in \mathbb{N}$). However, the optimal value for $c(F)$ remains unknown even for such graphs. Bollobás showed, using random regular graphs, that $c(F) \geq 1/2$ when $F$ contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of $c(K_{1,t,t}) \leq t$. We improve this to a uniform constant, showing $c(F) \leq 4$ for every almost bipartite graph $F$. This surprisingly makes the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_07222 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coloring graphs with forbidden almost bipartite subgraphs Anderson, James Bernshteyn, Anton Dhawan, Abhishek Combinatorics Alon, Krivelevich, and Sudakov conjectured in 1999 that for every finite graph $F$, there exists a quantity $c(F)$ such that $χ(G) \leq (c(F) + o(1)) Δ/ \logΔ$ whenever $G$ is an $F$-free graph of maximum degree $Δ$. The largest class of connected graphs $F$ for which this conjecture has been verified so far, by Alon, Krivelevich, and Sudakov themselves, comprises the almost bipartite graphs (i.e., subgraphs of the complete tripartite graph $K_{1,t,t}$ for some $t \in \mathbb{N}$). However, the optimal value for $c(F)$ remains unknown even for such graphs. Bollobás showed, using random regular graphs, that $c(F) \geq 1/2$ when $F$ contains a cycle. On the other hand, Davies, Kang, Pirot, and Sereni recently established an upper bound of $c(K_{1,t,t}) \leq t$. We improve this to a uniform constant, showing $c(F) \leq 4$ for every almost bipartite graph $F$. This surprisingly makes the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of our bound in the setting of DP-coloring (also known as correspondence coloring) and consider some algorithmic consequences of our results. |
| title | Coloring graphs with forbidden almost bipartite subgraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2203.07222 |