Bounds for the independence and chromatic numbers of locally sparse graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913950561468416 |
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| author | Dhawan, Abhishek |
| author_facet | Dhawan, Abhishek |
| contents | In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ cliques of size $r$. For $r \geq 3$ and $ε\in [0, 1]$, we show that an $n$-vertex $(Δ^{εr}, r)$-locally-sparse graph $G$ of maximum degree $Δ$ satisfies $α(G) \geq (1-o(1))\dfrac{n}{ηΔ}$ and $χ(G) \leq Θ\left(ηΔ\right)$, where $η:= ε+ \dfrac{r\log\log Δ}{\log Δ}$. For $ε$ not too large, the hidden constant in the $Θ(\cdot)$ can be taken to be $1+o(1)$. Setting $ε= 0$, we recover classical results on $K_{r+1}$-free graphs due to Shearer and Johansson, which were more recently improved by Davies, Kang, Pirot, and Sereni. We prove a stronger result on the independence number in terms of the occupancy fraction in the hard-core model, and establish a local version of the coloring result in the more general setting of correspondence coloring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for the independence and chromatic numbers of locally sparse graphs Dhawan, Abhishek Combinatorics In this note we consider a more general version of local sparsity introduced recently by Anderson, Kuchukova, and the author. In particular, we say a graph $G = (V, E)$ is $(k, r)$-locally-sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ cliques of size $r$. For $r \geq 3$ and $ε\in [0, 1]$, we show that an $n$-vertex $(Δ^{εr}, r)$-locally-sparse graph $G$ of maximum degree $Δ$ satisfies $α(G) \geq (1-o(1))\dfrac{n}{ηΔ}$ and $χ(G) \leq Θ\left(ηΔ\right)$, where $η:= ε+ \dfrac{r\log\log Δ}{\log Δ}$. For $ε$ not too large, the hidden constant in the $Θ(\cdot)$ can be taken to be $1+o(1)$. Setting $ε= 0$, we recover classical results on $K_{r+1}$-free graphs due to Shearer and Johansson, which were more recently improved by Davies, Kang, Pirot, and Sereni. We prove a stronger result on the independence number in terms of the occupancy fraction in the hard-core model, and establish a local version of the coloring result in the more general setting of correspondence coloring. |
| title | Bounds for the independence and chromatic numbers of locally sparse graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.03054 |