Bounds for the independence and chromatic numbers of locally sparse graphs

Abstract

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.