2026/05/04 by Ilkyoo Choi, Hojin Chu, Ringi Kim +1 · 1 voice
Mathematics · #math.CO
Dean conjectured that for each integer k ≥ 3, every graph with minimum degree at least k has a cycle whose length is divisible by k; this conjecture is known to be true for all k≠ 5. For k∈\3,4\, stronger statements are true: every graph with minimum degree at least 2 and at most k-2 vertices of degree 2 has a cycle whose length is divisible by k. We further strengthen these results by characterizing all graphs with minimum degree at least 2 and at most three vertices of degree 2 that have no cycle of length divisible by k, for each k∈\3,4\. As a corollary, we obtain that every graph with minimum degree at least 2 and at most two vertices of degree 2 has a cycle whose length is divisible by 3, and that every graph on at least nine vertices with minimum degree at least 2 and at most three vertices of degree 2 has a cycle whose length is divisible by 4.