Every Minimal Counterexample to the Erd\H{o}s-Gy\'arf\'as Conjecture is Predominantly Cubic
A minimal counterexample to the Erd\H{o}s-Gy\'arf\'as conjecture is a graph of minimum possible order and size with minimum degree at least 3 that contains no cycle whose length is a power of 2. Markstr\"om observed that any such graph must contain an independent set of vertices of degree at least 4 together with a nonempty set of vertices of degree exactly 3. As an immediate consequence, every regular minimal counterexample must be cubic. Building on this structure, two additional consequences are derived. First, every vertex of a minimal counterexample is adjacent to a vertex of degree exactly 3. Second, at least 4/7 of the vertices of any minimal counterexample must have degree exactly 3.
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