Coloring small locally sparse degenerate graphs and related problems
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915744629915648 |
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| author | Bradač, Domagoj Fox, Jacob Steiner, Raphael Sudakov, Benny Zhang, Shengtong |
| author_facet | Bradač, Domagoj Fox, Jacob Steiner, Raphael Sudakov, Benny Zhang, Shengtong |
| contents | The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{Ω(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$χ(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood.
For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lovász, Saks and Trotter from 1989.
Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15245 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coloring small locally sparse degenerate graphs and related problems Bradač, Domagoj Fox, Jacob Steiner, Raphael Sudakov, Benny Zhang, Shengtong Combinatorics 05C15, 05C83 The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{Ω(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$χ(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lovász, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research. |
| title | Coloring small locally sparse degenerate graphs and related problems |
| topic | Combinatorics 05C15, 05C83 |
| url | https://arxiv.org/abs/2601.15245 |