On the multicolor Turán conjecture for color-critical graphs

Abstract

A {\it simple $k$-coloring} of a multigraph $G$ is a decomposition of the edge multiset as a disjoint sum of $k$ simple graphs which are referred as colors. A subgraph $H$ of a multigraph $G$ is called {\it multicolored} if its edges receive distinct colors in a given simple $k$-coloring of $G$. In 2004, Keevash-Saks-Sudakov-Verstraëte introduced the {\it $k$-color Turán number} $e{x}_k(n,H)$, which denotes the maximum number of edges in an $n$-vertex multigraph that has a simple $k$-coloring containing no multicolored copies of $H$. They made a conjecture for any $r\ge 3$ and $r$-color-critical graph $H$ that in the range of $k\ge \frac{r-1}{r-2}(e(H)-1)$, if $n$ is sufficiently large, then $e{x}_k(n,H)$ is achieved by the multigraph consisting of $k$ colors all of which are identical copies of the Turán graph $T_{r-1}(n)$. In this paper, we show that this holds in the range of $k\ge 2\frac{r-1}{r}(e(H)-1)$, significantly improving earlier results. Our proof combines the stability argument of Chakraborti-Kim-Lee-Liu-Seo with a novel graph packing technique for embedding multigraphs.

29 pages, accepted by Canadian Journal of Mathematics