Exact solutions to the Erd\H{o}s-Rothschild problem

TL;DR

This paper provides new exact solutions to the Erd ext{"o}s-Rothschild problem, identifying extremal graphs for specific parameters and conditions, including cases involving complete multipartite graphs and Hadamard matrices.

Contribution

It establishes conditions for extremal graphs to be complete multipartite and solves specific cases of the Erd ext{"o}s-Rothschild problem for certain parameters.

Findings

Extremal graphs are complete multipartite under certain conditions.

Unique extremal graph for ,,...,3 (length 7) is an 8-partite graph derived from Hadamard matrices.

For ,,...,k with 3  k  10, extremal graphs are nearly balanced k-partite graphs.

Abstract

Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\textbf{k})$ to denote the maximum of $F(G;\textbf{k})$ over all graphs $G$ on $n$ vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erd\H{o}s and Rothschild in 1974. We prove some new exact results for $n \to \infty$: (i) A sufficient condition on $\textbf{k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erd\H{o}s and Rothschild, in the case $\textbf{k}=(3,\ldots,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$. (iii) In the case $\textbf{k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is complete $k$-partite with one part of size less than $k$ and the other parts as equal in size as possible.