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

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 $d_1\le d_2\le \cdots \le d_n$ 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 ${\log }_{2}n$.