An edge coloring of a graph G is a Gallai coloring if it contains no rainbow triangle. We show
that the number of Gallai r-colorings of K_n is (r2+o(1))2^n2. This result indicates that almost …

Given a graph F and an integer r≥ 2, a partition F ̂ of the edge set of F into at most r
classes, and a graph G, define cr, F ̂ (G) as the number of r-colorings of the edges of G that …

An edge coloring of the n-vertex complete graph, K n, is a Gallai coloring if it does not
contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct …

A {\it simple $ k $-coloring} of a multigraph $ G $ is a decomposition of the edge multiset as a
disjoint sum of $ k $ simple graphs which are referred as colors. A subgraph $ H $ of a …

We consider a multicolored version of a question posed by Erdös and Rothschild. For a
fixed positive integer r and a fixed graph F, we look for n-vertex graphs that admit the …

Inspired by previous work of Balogh (2006), we show that, given r≥ 5 and n large, the
balanced complete bipartite graph K n∕ 2, n∕ 2 is the n-vertex graph that admits the largest …

An edge coloring of the n-vertex complete graph K n is a Gallai coloring if it does not contain
any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors …

For positive integers n and r, we consider n-vertex graphs with the maximum number of r-
edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if …

Given integers r≥ 2, k≥ 3 and 2≤ s≤(k 2), and a graph G, we consider r-edge-colorings of
G with no copy of a complete graph K k on k vertices where s or more colors appear, which …

We consider a version of the Erdős–Rothschild problem for families of graph patterns. For
any fixed k≥ 3, let r 0 (k) be the largest integer such that the following holds for all 2≤ r≤ r 0 …