Erdős-Gyárfás Balanced Five-Colorings

Abstract

A coloring of the edges of a complete graph with $r$ colors is balanced if the edges among any $r+1$ vertices use all $r$ colors. Erdős and Gyárfás (1999) conjectured that for all $r\ge3$ the complete graph $K_{r^2+1}$ has no balanced $r$-coloring. They proved the conjecture for $r=3$ and $r=4$. We prove the conjecture for $r=5$, the first new case settled in over two decades. There is no balanced $5$-coloring of $K_{26}$, and since $K_{25}$ has one, $K_{25}$ is the largest complete graph admitting a balanced $5$-coloring. We design a reduce--dispatch--enumerate workflow, where every machine step produces an artifact that can be independently checked. First, we use a machine-checked (Lean 4) proof to reduce the conjecture to a single extremal statement about the color class with the fewest edges: no graph on $26$ vertices has between $55$ and $65$ edges while every $6$ of its vertices span between $1$ and $11$ edges. We then search on the sparse class instead of its dense complement, which makes enumeration up to isomorphism feasible. With classical stability theorems for $K_6$-free graphs, we dispatch the five near-extremal edge counts analytically. For the exhaustive enumeration on the remaining six edge counts, we design efficient encodings with SAT modulo symmetries and parallel cube-and-conquer. We also certify all eleven edge counts: each of the $12{,}296$ subproblems carries a clausal refutation checked by the independent verifier drat-trim, and all $273{,}503{,}938$ symmetry-breaking clauses are also validated against the vertex permutations that justify them.