Making an H $H$‐free graph k $k$‐colorable

Abstract

We study the following question: How few edges can we delete from any H $H$‐free graph on n $n$ vertices to make the resulting graph k $k$‐colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For H $H$ any fixed odd cycle, we determine the answer up to a constant factor when n $n$ is sufficiently large. We also prove an upper bound when H $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges.

New results for MaxCut in H$H$‐free graphs

“…By (1.4), these graphs have surplus at most 𝑂(𝜆𝑛) = 𝑂((𝑛𝑑) (𝑟+1)∕(𝑟+2) ), which shows that (1.3) would be optimal for all odd 𝑟. Regarding this problem, Zeng and Hou [24] showed that sp(𝑚, 𝐶 𝑟 ) ⩾ 𝑚 (𝑟+1)∕(𝑟+3)+𝑜 (1) for all odd 𝑟, and very recently, Fox, Himwich and Mani [14] improved the surplus to Ω 𝑟 (𝑚 (𝑟+5)∕(𝑟+7) ). We settle this problem completely by proving the following tight result, which confirms the conjecture of Alon, Krivelevich and Sudakov [5].…”

“…The study of MaxCut in $$H$$‐free graphs was initiated by Erdős and Lovász (see [12]) in the 70s, and has received significant attention since then (e.g. [2, 3, 5, 6, 14, 21, 22, 24]). For a graph $$H$$, define $${\rm sp}(m,H)$$ as the minimum surplus $${\rm sp}(G)={\rm mc}(G)-m/2$$ over all $$H$$‐free graphs $$G$$ with $$m$$ edges.…”

“…By (), these graphs have surplus at most $$O(\lambda n)=O((nd)^{(r+1)/(r+2)})$$, which shows that () would be optimal for all odd $$r$$. Regarding this problem, Zeng and Hou [24] showed that $${\rm sp}(m,C_r)\geqslant m^{(r+1)/(r+3)+o(1)}$$ for all odd $$r$$, and very recently, Fox, Himwich and Mani [14] improved the surplus to $$\Omega _r(m^{(r+5)/(r+7)})$$. We settle this problem completely by proving the following tight result, which confirms the conjecture of Alon, Krivelevich and Sudakov [5].…”