What question did this study set out to answer?

This work aims to establish stronger structural bounds for vertices in minimal counterexamples to the Erdős–Gyárfás conjecture.

July 27, 2026Open Access

Two Thirds of the Vertices of a Minimal Counterexample to the Erdős–Gyárfás Conjecture are Cubic

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Randomized trial uncovers structural properties of cubic vertices in minimal counterexamples, suggesting deeper insights into graph theory.

This work aims to establish stronger structural bounds for vertices in minimal counterexamples to the Erdős–Gyárfás conjecture.
Combining structural insights from previous works
Establishing new restrictions based on vertex connections
Analyzing degree distributions within the graph structure
Demonstrated that at least 2/3 of vertices in counterexamples have degree exactly 3
Proved a unique common neighbor condition for pairs of adjacent cubic vertices
Showed that adjacent cubic vertices with specific edge connections yield a neighbor with degree exactly 4

Andrew Bisch (2026) studied this question.

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Randomized trial uncovers structural properties of cubic vertices in minimal counterexamples, suggesting deeper insights into graph theory.