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A Gallai-coloring (Gallai-$k$-coloring) is an edge-coloring (with colors from ${1, 2, ldots, k}$) of a complete graph without rainbow triangles. Given a graph $H$ and a positive integer $k$, the $k$-colored Gallai-Ramsey number $GR_k(H)$ is the minimum integer $n$ such that every Gallai-$k$-coloring of the complete graph $K_n$ contains a monochromatic copy of $H$. In this paper, we consider two extremal problems related to Gallai-$k$-colorings. First, we determine upper and lower bounds for the maximum number of edges that are not contained in any rainbow triangle or monochromatic triangle in a $k$-edge-coloring of $K_n$. Second, for $ngeq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, yielding the exact value for $k=3$. Furthermore, we determine the Gallai-Ramsey number $GR_k(K_4+e)$ for the graph on five vertices consisting of a $K_4$ with a pendant edge.
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We survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries.
A Gallai coloring is an edge coloring that avoids triangles colored with three different colors. Given integers $e_1ge e_2 ge dots ge e_k$ with $sum_{i=1}^ke_i={n choose 2}$ for some $n$, does there exist a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$? In this paper, we give several sufficient conditions and one necessary condition to guarantee a positive answer to the above question. In particular, we prove the existence of a Gallai-coloring if $e_1-e_kle 1$ and $k le lfloor n/2rfloor$. We prove that for any integer $kge 3$ there is a (unique) integer $g(k)$ with the following property: there exists a Gallai $k$-coloring of $K_n$ with $e_i$ edges in color $i$ for every $e_1ledots le e_k$ satisfying $sum_{i=1}^ke_i={nchoose 2}$, if and only if $nge g(k)$. We show that $g(3)=5$, $g(4)=8$, and $2k-2le g(k)le 8k^2+1$ for every $kge 3$.
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.
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 $2leq qleq lfloorlog_2 (p-1)rfloor+1$.
In a generalized Turan problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
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