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Register a new userExact results on generalized ErdH{o}s-Gallai problems
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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.
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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.
Extending the concept of Ramsey numbers, Erd{H o}s and Rogers introduced the following function. For given integers $2le s<t$ let $$ f_{s,t}(n)=min {max {|W| : Wsubseteq V(G) {and} G[W] {contains no} K_s} }, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this paper, we show that for every $sge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) le c_1 (log n)^{c_2} sqrt{n}$. This result is best possible up to a polylogarithmic factor. We also show for all $t-2 geq s geq 4$, there exists a constant $c_3$ such that $f_{s,t}(n) le c_3 sqrt{n}$. In doing so, we partially answer a question of ErdH{o}s by showing that $lim_{nto infty} frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)}=infty$ for any $sge 4$.
For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ is the minimum number of colors needed for $K_n$ to have a $(p, q)$-coloring. This function was introduced about 45 years ago, but was studied systematically by ErdH{o}s and Gy{a}rf{a}s in 1997, and is now known as the ErdH{o}s-Gy{a}rf{a}s function. In this paper, we study $f(n, p, q)$ with respect to Gallai-colorings, where a Gallai-coloring is an edge-coloring of $K_n$ without rainbow triangles. Combining the two concepts, we consider the function $g(n, p, q)$ that is the minimum number of colors needed for a Gallai-$(p, q)$-coloring of $K_n$. Using the anti-Ramsey number for $K_3$, we have that $g(n, p, q)$ is nontrivial only for $2leq qleq p-1$. We give a general lower bound for this function and we study how this function falls off from being equal to $n-1$ when $q=p-1$ and $pgeq 4$ to being $Theta(log n)$ when $q = 2$. In particular, for appropriate $p$ and $n$, we prove that $g=n-c$ when $q=p-c$ and $cin {1,2}$, $g$ is at most a fractional power of $n$ when $q=lfloorsqrt{p-1}rfloor$, and $g$ is logarithmic in $n$ when $2leq qleq lfloorlog_2 (p-1)rfloor+1$.
For a 2-connected graph $G$ on $n$ vertices and two vertices $x,yin V(G)$, we prove that there is an $(x,y)$-path of length at least $k$ if there are at least $frac{n-1}{2}$ vertices in $V(G)backslash {x,y}$ of degree at least $k$. This strengthens a well-known theorem due to ErdH{o}s and Gallai in 1959. As the first application of this result, we show that a 2-connected graph with $n$ vertices contains a cycle of length at least $2k$ if it has at least $frac{n}{2}+k$ vertices of degree at least $k$. This confirms a 1975 conjecture made by Woodall. As another applications, we obtain some results which generalize previous theorems of Dirac, ErdH{o}s-Gallai, Bondy, and Fujisawa et al., present short proofs of the path case of Loebl-Koml{o}s-S{o}s Conjecture which was verified by Bazgan et al. and of a conjecture of Bondy on longest cycles (for large graphs) which was confirmed by Fraisse and Fournier, and make progress on a conjecture of Bermond.
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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