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To prove the five-color case of the Erdős–Gyárfás conjecture regarding edge colorings of complete graphs.

July 23, 2026Open Access

The Five-Color Case of an Erdős–gyárfás Balanced-Coloring Problem

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Randomized proof establishes the five-color case of a conjecture in graph theory, indicating a significant property of edge colorings.

To prove the five-color case of the Erdős–Gyárfás conjecture regarding edge colorings of complete graphs.
Proof applies the Kang–Pikhurko extremal theorem to complements of induced subgraphs.
Assignments of graphs for each color are utilized to derive edge bounds with minimum-degree decompositions.
Near-extremal lemmas are employed to force conditions on the edge counts of five color graphs.
The proof demonstrates that every five-coloring of edges in K26 has six vertices with omitted colors.
Establishes that the set-coloring Ramsey number R(6; 5, 4) equals 26.
Confirms that five-color graphs possess at least 65 edges, excluding equality cases.

Robert Sneiderman (2026) studied this question.

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