Independent sets and colorings of $K_{t,t,t}$-free graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866917123683516416 |
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| author | Dhawan, Abhishek Janzer, Oliver Methuku, Abhishek |
| author_facet | Dhawan, Abhishek Janzer, Oliver Methuku, Abhishek |
| contents | Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $Δ$ has chromatic number $O(Δ/ \log Δ)$. This was previously known only for almost bipartite graphs, that is, for subgraphs of $K_{1,t,t}$ (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where $F$ is almost bipartite. We prove this conjecture for all $3$-colorable graphs $F$, i.e. subgraphs of $K_{t,t,t}$, representing the first progress toward the conjecture since it was posed.
A closely related conjecture of Ajtai, Erdős, Komlós, and Szemerédi from 1981 asserts that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $Ω(n \log d / d)$. We prove this conjecture in a strong form for all 3-colorable graphs $F$. More precisely, we show that every $n$-vertex $K_{t,t,t}$-free graph of average degree $d$ contains an independent set of size at least $(1 - o(1)) n \log d / d$, matching Shearer's celebrated bound for triangle-free graphs (the case $t = 1$) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the Rödl nibble method for constructing independent sets with a Turán-type result on $K_{t,t,t}$-free graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_17191 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Independent sets and colorings of $K_{t,t,t}$-free graphs Dhawan, Abhishek Janzer, Oliver Methuku, Abhishek Combinatorics Discrete Mathematics Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $Δ$ has chromatic number $O(Δ/ \log Δ)$. This was previously known only for almost bipartite graphs, that is, for subgraphs of $K_{1,t,t}$ (verified by Alon, Krivelevich, and Sudakov themselves), while most recent results were concerned with improving the leading constant factor in the case where $F$ is almost bipartite. We prove this conjecture for all $3$-colorable graphs $F$, i.e. subgraphs of $K_{t,t,t}$, representing the first progress toward the conjecture since it was posed. A closely related conjecture of Ajtai, Erdős, Komlós, and Szemerédi from 1981 asserts that for every graph $F$, every $n$-vertex $F$-free graph of average degree $d$ contains an independent set of size $Ω(n \log d / d)$. We prove this conjecture in a strong form for all 3-colorable graphs $F$. More precisely, we show that every $n$-vertex $K_{t,t,t}$-free graph of average degree $d$ contains an independent set of size at least $(1 - o(1)) n \log d / d$, matching Shearer's celebrated bound for triangle-free graphs (the case $t = 1$) and thereby yielding a substantial strengthening of it. Our proof combines a new variant of the Rödl nibble method for constructing independent sets with a Turán-type result on $K_{t,t,t}$-free graphs. |
| title | Independent sets and colorings of $K_{t,t,t}$-free graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2511.17191 |