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On the multicolor Turán conjecture for color-critical graphs
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 10 September 2025, pp. 1-31
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- Article
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A 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 to as colors. A subgraph H of a multigraph G is called multicolored if its edges receive distinct colors in a given simple k-coloring of G. In 2004, Keevash–Saks–Sudakov–Verstraëte introduced the k-color Turán number
${\text {ex}}_k(n,H)$, 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 
$r\geq 3$ and r-color-critical graph 
$H,$ that in the range of 
$k\geq \frac {r-1}{r-2}(e(H)-1)$, if n is sufficiently large, then 
${\text {ex}}_k(n, H)$ 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 article, we show that this holds in the range of 
$k\geq 2\frac {r-1}{r}(e(H)-1)$, 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.
