Full Characterization of Color Degree Sequences in Complete Graphs Without Tricolored Triangles

Abstract

For an edge-colored complete graph, we define the color degree of a node as the number of colors appearing on edges incident to it. In this paper, we consider colorings that don't contain tricolored triangles (also called rainbow triangles); these colorings are also called Gallai colorings. We give a complete characterization of all possible color degree sequences $d_1 \le d_2 \le \dots \le d_n$ that can arise on a Gallai coloring of $K_n$: it is necessary and sufficient that \[ \sum_{i = k}^n \frac{1}{2^{d_i - d_{k-1}}} \ge 1 \] holds for all $1 \le k \le n$, where $d_0=0$ for convenience. As a corollary, this gives another proof of a 2018 result of Fujita, Li, and Zhang who showed that the minimum color degree in such a coloring is at most $\log_2n$.