An extension of Dirac’s Theorem

The following extension of Dirac’s Theorem was conjectured by Enomoto, Kaneko and Tuza: if G is an n-vertex graph with minimum degree at least n/k, then there are k − 1 cycles in G covering the ∗Research is partially supported by Simons Fellowship, NSF CAREER Grant DMS0745185, Marie Curie FP7-PEOPLE-2012-IIF 327763. †Supported by grant no. 6910960 of the Fonds National de la Recherche, Luxembourg. vertex set V (G). Kouider proved this conjecture under the extra condition that G is 2-connected, and pointed out that this condition is necessary when k > √ n. Here, we show that, for a fixed integer k and n sufficiently large, if G is an n-vertex graph with minimum degree at least n/k, then there are k−1 cycles in G covering the vertex set V (G). This bound is best possible since there exist graphs with minimum degree n/k − 1 which do not have this property. In our proof we use modern methods, including using the Szemerédi Regularity Lemma and the technique of ‘connected matchings’. These methods also allow us to give a simple description of the extremal structure of graphs with minimum degree almost n/k that cannot be covered with k − 1 cycles.