On cuts of small chromatic number in sparse graphs
Guillaume Aubian, Marthe Bonamy, Romain Bourneuf, Oscar Fontaine, Lucas Picasarri-Arrieta
TL;DR
The paper investigates the threshold $\ell_k$ governing when large sparse graphs admit a separator $X$ with $\chi(G[X])<k$. It disproves the natural conjecture $\ell_k=k$ by constructing, via a layered clique-based graph built around a bipartite base $H$ with no large bi-holes (guaranteed by a result of ASSW21), graphs of average degree about $2\ell$ in which every separator induces a subgraph with $\chi(G[X])\ge k$, thus forcing $\ell_k \le (1+\varepsilon)\tfrac{k}{2}$ for any fixed $\varepsilon>0$ and large $k$. The key idea is to force any cut to reveal a bi-hole in the bipartite base, which then implies a large chromatic number in the separator due to the clique structure within blocks. The authors provide quantitative parameter choices showing $\ell_k$ is asymptotically $\tfrac{k}{2}$ and that there exist arbitrarily large graphs with average degree $(1+o(1))k$ whose separators must contain a $k$-clique, establishing essentially tight bounds and answering a central question about cuts in sparse graphs. This work clarifies the limits of the conjectured relation between average degree and chromatic-constraint separators and motivates further study of the threshold for small $k$ and possible stronger degeneracy-based variants.
Abstract
For a given integer $k$, let $\ell_k$ denote the supremum $\ell$ such that every sufficiently large graph $G$ with average degree less than $2\ell$ admits a separator $X \subseteq V(G)$ for which $χ(G[X]) < k$. Motivated by the values of $\ell_1$, $\ell_2$ and $\ell_3$, a natural conjecture suggests that $\ell_k = k$ for all $k$. We prove that this conjecture fails dramatically: asymptotically, the trivial lower bound $\ell_k \geq \tfrac{k}{2}$ is tight. More precisely, we prove that for every $\varepsilon>0$ and all sufficiently large $k$, we have $\ell_k \leq (1+\varepsilon)\tfrac{k}{2}$.