Minimum degree conditions for cycles

In this note we discuss the lengths of cycles which are forced to exist in an n-vertex graph with minimum degree δ. We show that for any integer k ≥ 2 there exists n0 such that if n ≥ n0 and G is an n-vertex graph with δ(G) = δ ≥ n/k then the following are true. (i) G contains Ct for every even 4 ≤ t ≤ ⌈ n k−1 ⌉ , (ii) either G is in a known exceptional class or G contains Ct for every odd t ∈ [ ⌊ 2n δ ⌋− 1, δ + 1], and (iii) if G does not contain a cycle of every length from ⌊ 2n δ ⌋− 1 to ⌈ n k−1 ⌉ inclusive then G does contain Ct for every even 4 ≤ t ≤ 2δ. This is an improvement on a theorem of Nikiforov and Schelp [7]. We recall that the circumference c(G) of a graph G is the length of the longest cycle in G; we define also oc(G) and ec(G) to be the lengths of the longest odd and even cycles in G. Early results of Dirac [3] and Voss and Zuluaga [8] examined 2-connected graphs: Theorem 1. (Dirac [3]) If G is a 2-connected graph with minimum degree δ then c(G) ≥ min(|V (G)|, 2δ). Theorem 2. (Voss and Zuluaga [8]) If G is a 2-connected graph on at least 2δ vertices with minimum degree at least δ then ec(G) ≥ 2δ; furthermore if G is not bipartite then also oc(G) ≥ 2δ − 1. More recently various authors [1, 2, 4] removed the connectivity requirement, resulting in: ∗email: [email protected]