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Register a new userExact solutions to the ErdH{o}s-Rothschild problem
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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 known for this problem, posed by ErdH{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 ErdH{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.
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Given a sequence $mathbf{k} := (k_1,ldots,k_s)$ of natural numbers and a graph $G$, let $F(G;mathbf{k})$ 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 $F(n;mathbf{k})$ to denote the maximum of $F(G;mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by ErdH{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. In previous work with Yilma, we constructed a finite optimisation problem whose maximum is equal to the limit of $log_2 F(n;mathbf{k})/{nchoose 2}$ as $n$ tends to infinity and proved a stability theorem for complete multipartite graphs $G$. In this paper we provide a sufficient condition on $mathbf{k}$ which guarantees a general stability theorem for any graph $G$, describing the asymptotic structure of $G$ on $n$ vertices with $F(G;mathbf{k}) = F(n;mathbf{k}) cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$. The proof uses a novel version of symmetrisation on edge-coloured weighted multigraphs.
Let $mathbf{k} := (k_1,dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;mathbf{k})$ 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 $F(n;mathbf{k})$ to denote the maximum of $F(G;mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by ErdH{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;mathbf{k}) = F(n;mathbf{k})$. Also, for every $mathbf{k}$ we construct a finite optimisation problem whose maximum is equal to the limit of $log_2 F(n;mathbf{k})/{nchoose 2}$ as $n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;mathbf{k}) = F(n;mathbf{k}) cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.
The triangle covering number of a graph is the minimum number of vertices that hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph $G$ with $lfloor n^2/4 rfloor +t$ edges and triangle covering number $s$, we determine (for large $n$) sharp bounds on the minimum number of triangles in $G$ and also describe the extremal constructions. Similar results are proved for cliques of larger size and color critical graphs. This extends classical work of Rademacher, ErdH os, and Lovasz-Simonovits whose results apply only to $s le t$. Our results also address two conjectures of Xiao and Katona. We prove one of them and give a counterexample and prove a modified version of the other conjecture.
Generalized Turan problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain any path (or cycle) of length at least a given number. Both of the path-free and cycle-free extremal problems were recently considered and asymptotically solved by Luo. We fully resolve these problems by characterizing all possible extremal graphs. We further extend these results by solving the edge-variant of these problems where the number of edges is fixed instead of the number of vertices. We similarly obtain exact characterization of the extremal graphs for these edge variants.
In 1981, ErdH{o}s and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $mathcal{C}_text{odd}(G)$ be the set of odd numbers in $mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $sum_{ellin mathcal{C}_text{odd}(G)}1/ellgeq (1/2-o_k(1))log k$. This solves ErdH{o}s and Hajnals odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, ErdH{o}s asked whether there is some $d$ such that each graph with chromatic number at least $d$ (or perhaps even only average degree at least $d$) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every $k$, there is some $d$ so that every graph with average degree at least $d$ has a subdivision of the complete graph $K_k$ in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.
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