Local Shearer bound
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917882417381376 |
|---|---|
| author | Martinsson, Anders Steiner, Raphael |
| author_facet | Martinsson, Anders Steiner, Raphael |
| contents | We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every vertex $v$ of $G$ is contained in a random independent set drawn from the distribution with probability $(1-o(1))\frac{\ln d(v)}{d(v)}$. This resolves the main conjecture raised by Kelly and Postle (2018) about fractional coloring with local demands, which in turn confirms a conjecture by Cames van Batenburg et al. (2018) stating that every $n$-vertex triangle-free graph has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{\frac{n}{\ln(n)}}$. Addressing another conjecture posed by Cames van Batenburg et al., we also establish an analogous upper bound in terms of the number of edges.
To prove these results we establish a more general technical theorem that works in a weighted setting. As a further application of this more general result, we obtain a new spectral upper bound on the fractional chromatic number of triangle-free graphs: We show that every triangle-free graph $G$ satisfies $χ_f(G)\le (1+o(1))\frac{ρ(G)}{\ln ρ(G)}$ where $ρ(G)$ denotes the spectral radius. This improves the bound implied by Wilf's classic spectral estimate for the chromatic number by a $\ln ρ(G)$ factor and makes progress towards a conjecture of Harris on fractional coloring of degenerate graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00567 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Shearer bound Martinsson, Anders Steiner, Raphael Combinatorics 05C07, 05C15, 05C69, 05C72 We prove the following local strengthening of Shearer's classic bound on the independence number of triangle-free graphs: For every triangle-free graph $G$ there exists a probability distribution on its independent sets such that every vertex $v$ of $G$ is contained in a random independent set drawn from the distribution with probability $(1-o(1))\frac{\ln d(v)}{d(v)}$. This resolves the main conjecture raised by Kelly and Postle (2018) about fractional coloring with local demands, which in turn confirms a conjecture by Cames van Batenburg et al. (2018) stating that every $n$-vertex triangle-free graph has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{\frac{n}{\ln(n)}}$. Addressing another conjecture posed by Cames van Batenburg et al., we also establish an analogous upper bound in terms of the number of edges. To prove these results we establish a more general technical theorem that works in a weighted setting. As a further application of this more general result, we obtain a new spectral upper bound on the fractional chromatic number of triangle-free graphs: We show that every triangle-free graph $G$ satisfies $χ_f(G)\le (1+o(1))\frac{ρ(G)}{\ln ρ(G)}$ where $ρ(G)$ denotes the spectral radius. This improves the bound implied by Wilf's classic spectral estimate for the chromatic number by a $\ln ρ(G)$ factor and makes progress towards a conjecture of Harris on fractional coloring of degenerate graphs. |
| title | Local Shearer bound |
| topic | Combinatorics 05C07, 05C15, 05C69, 05C72 |
| url | https://arxiv.org/abs/2501.00567 |