Gallai-Ramsey numbers for multiple triangles
In this paper, the authors consider finite, undirected graphs without loops or parallel edges. Let \(K_n\) be the complete graph on \(n\) vertices. A Gallai coloring of \(K_n\) is an edge-coloring, such that any triangle of \(K_n\) contains two edges with the same color. A Gallai coloring that uses at most \(k\) colors is called a Gallai \(k\)-coloring. For a positive integer \(k>1\) and graphs \(H_1,\dots,H_k\) let \(GR(H_1,\dots,H_k)\) be the smallest number \(N\) such that for every Gallai \(k\)-coloring of \(K_N\) there is \(i\in \{1,\dots,k\}\) and a monochromatic copy of \(H_i\) in \(K_N\). For a positive integer \(m\) let \(mG\) be a graph obtained from disjoint union of \(m\) copies of \(G\). Within the frames of this paper, the authors study the number \(GR(t_1K_3,\dots,t_kK_3)\) for \(t_1\geq\dots\geq t_k\geq 1\). They obtain upper and lower bounds for \(GR(t_1K_3,\dots,t_kK_3)\) with \(t_1\geq \dots\geq t_k\geq 1\), which they prove to be tight for some \(t_i\) with \(i\in \{1,\dots,k\}\).
scientific article; zbMATH DE number 7352777
scientific article