On the chromatic number of random triangle-free graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911132253421568 |
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| author | Mizgerd, Clayton Perkins, Will Wang, Yuzhou |
| author_facet | Mizgerd, Clayton Perkins, Will Wang, Yuzhou |
| contents | We study the chromatic number of typical triangle-free graphs with $Θ\left( n^{3/2} (\log n)^{1/2} \right)$ edges and establish the width of the scaling window for the transitions from $χ= 3$ to $χ= 4$ and from $χ= 4$ to $χ= 5$.
The transition from $3$- to $4$-colorability has scaling window of width $Θ(n^{4/3} (\log n)^{-1/3})$. To prove this, we show a high probability equivalence of the $3$-colorability of a random triangle-free graph at this density and the satisfiability of an instance of bipartite random $2$-SAT, for which we establish the width of the scaling window following the techniques of Bollob{á}s, Borgs, Chayes, Kim, and Wilson.
The transition from $4$- to $5$-colorability has scaling window of width $Θ(n^{3/2} (\log n)^{-1/2})$. To prove this, we show a high probability equivalence of the $4$-colorability of a random triangle-free graph at this density and the simultaneous $2$-colorability of two independent Erdős--Rényi random graphs. For this transition, we also establish the limiting probability of $4$-colorability inside the scaling window. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the chromatic number of random triangle-free graphs Mizgerd, Clayton Perkins, Will Wang, Yuzhou Combinatorics Probability We study the chromatic number of typical triangle-free graphs with $Θ\left( n^{3/2} (\log n)^{1/2} \right)$ edges and establish the width of the scaling window for the transitions from $χ= 3$ to $χ= 4$ and from $χ= 4$ to $χ= 5$. The transition from $3$- to $4$-colorability has scaling window of width $Θ(n^{4/3} (\log n)^{-1/3})$. To prove this, we show a high probability equivalence of the $3$-colorability of a random triangle-free graph at this density and the satisfiability of an instance of bipartite random $2$-SAT, for which we establish the width of the scaling window following the techniques of Bollob{á}s, Borgs, Chayes, Kim, and Wilson. The transition from $4$- to $5$-colorability has scaling window of width $Θ(n^{3/2} (\log n)^{-1/2})$. To prove this, we show a high probability equivalence of the $4$-colorability of a random triangle-free graph at this density and the simultaneous $2$-colorability of two independent Erdős--Rényi random graphs. For this transition, we also establish the limiting probability of $4$-colorability inside the scaling window. |
| title | On the chromatic number of random triangle-free graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2509.01376 |