Coloring locally sparse graphs

A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $Δ$ is $Δ^2/f$-locally-sparse, then $χ(G) = O\left(Δ/\log f\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph $F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the Rödl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $χ(G) = O\left( Δ/\log\left(Δk^{-1/|V(F)|}\right)\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $χ(G)$ when $G$ is $K_{1, t, t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the more general so-called ‘‘color-degree’’ setting.

💡 Research Summary

The paper revisits the classical notion of local sparsity in graphs—where a graph is called k‑locally‑sparse if each vertex’s neighbourhood induces at most k edges—and extends it to a far more flexible framework. For a fixed small graph F, a graph G is defined to be (k,F)‑locally‑sparse if for every vertex v the induced subgraph on its neighbourhood contains at most k (not necessarily vertex‑disjoint) copies of F. This generalization captures the original case (F = K₂) as well as the recent work on paths (F = Pₜ) and many other forbidden substructures.

The authors prove that for any bipartite graph F and any sufficiently large maximum degree Δ, if G is (k,F)‑locally‑sparse with \