Extremal problems and results related to Gallai-colorings

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 prove that for any positive integers $d$ and $k$, there exists a constant $c$ such that if $H$ is an $n$-vertex graph with maximum degree $d$, then $GR_k(H)$ is at most $cn$. We also determine $GR_k(K_4+e)$ for the graph on 5 vertices consisting of a $K_4$ with a pendant edge. Furthermore, we consider two extremal problems related to Gallai-$k$-colorings. For $n\geq GR_k(K_3)$, we determine upper and lower bounds for the minimum number of monochromatic triangles in a Gallai-$k$-coloring of $K_{n}$, implying that this number is $O(n^3)$ and yielding the exact value for $k=3$. We also 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 $K_n$.Comment: 20 pages, 1 figur