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On a rainbow extremal problem for color-critical graphs

Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Hong Liu, Jaehyeon Seo

Abstract

There has been extensive studies on the following question: given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a `colorful' copy of $H$, i.e., a copy of $H$ containing at most one edge from each $G_i$? A lower bound on $\sum_{i\in [k]} e(G_i)$ enforcing a colorful copy of a given graph $H$ was considered by Keevash, Saks, Sudakov, and Verstraëte. They defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs $G_1,\dots, G_k$ on a common vertex set of size $n$ having no colorful copy of $H$. They completely determined $\operatorname{ex}_k(n,K_r)$ for large $n$ by showing that, depending on the value of $k$, one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for 3-color-critical graphs. We prove their conjecture for 4-color-critical graphs and for almost all $r$-color-critical graphs when $r > 4$. Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of $k$. This answers a question of Keevash, Saks, Sudakov, and Verstraëte.

On a rainbow extremal problem for color-critical graphs

Abstract

There has been extensive studies on the following question: given graphs over a common vertex set of size , what conditions on ensures a `colorful' copy of , i.e., a copy of containing at most one edge from each ? A lower bound on enforcing a colorful copy of a given graph was considered by Keevash, Saks, Sudakov, and Verstraëte. They defined to be the maximum total number of edges of the graphs on a common vertex set of size having no colorful copy of . They completely determined for large by showing that, depending on the value of , one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for 3-color-critical graphs. We prove their conjecture for 4-color-critical graphs and for almost all -color-critical graphs when . Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of . This answers a question of Keevash, Saks, Sudakov, and Verstraëte.
Paper Structure (20 sections, 26 theorems, 68 equations, 1 figure)

This paper contains 20 sections, 26 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization.
  3. Construction for non-color-critical graphs
  4. Preliminaries and necessary tools
  5. Notations
  6. Goodness and F
  7. Minimum degree condition of the host graph
  8. Nested colorings
  9. The condition for embedding a multicolored multigraph
  10. H-friendly submultigraphs
  11. Ignoring large k
  12. 4-vertex 4-color-critical multigraphs are good
  13. r-vertex multigraphs in F are good
  14. Stability for r-vertex r-color-critical multigraphs
  15. Stability for r-color-critical graphs on more than r vertices
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose that $r \ge 2$, $k \ge \binom{r}{2}$, and $n > 10^4r^{34}$. Let $G$ be an $n$-vertex $k$-color extremal multigraph of $K_r$. Then either all colors of $G$ are identical Turán graphs $T_{r-1}(n)$, or there are exactly $\binom{r}{2} - 1$ non-empty colors of $G$, all of which are complete graph

Figures (1)

  • Figure 1: The labelings of $G_0$ and $H$.

Theorems & Definitions (47)

  • Theorem 1.1: keevash2004multicolour
  • Conjecture 1.2: keevash2004multicolour
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.6
  • proof : Proof of \ref{['thm:ex-gph_non-cc-gph']}
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 37 more