Longest cycles in regular graphs (Q1070244)

Let c(G) denote the length of a longest cycle in a (simple) graph G. The well-known result of \textit{G. A. Dirac} [Ann. Discrete Math. 3, 75-92 (1978; Zbl 0376.05035)] states that if G is a 2-connected graph on n vertices with minimum degree k, then c(G)\(\geq \min (2k,n)\). This result was improved by \textit{B. Bollobas} and \textit{A. M. Hobbs} [Ann. Discrete Math. 3, 43-48 (1978; Zbl 0376.05036)] and \textit{R. Jackson} [J. Comb. Theory, Ser. B 29, 27-46 (1980; Zbl 0377.05027)] by adding regularity conditions. Here, the author improves these results for graphs of sufficiently small order by establishing the following. Theorem 1. Let G be a 3-connected, k-regular graph with n vertices. Then c(G)\(\geq \min (3k,n)\). Theorem 2. Let G be a 2-connected, k-regular graph with n vertices. If \(n<3k+4\), then c(G)\(\geq \min (3k,n)\).