On Longest Cycles in 2-connected Graphs - Mossine S. Koulakzian; Zhora G. Nikoghosyan (Изд-во НАН РА)
For a graph G, n denotes the order (the number of vertices) of G, c the order of a longest cycle in G (called circumference), p the order of a longest path and ± the minimum degree. In 1952, Dirac proved: (i) if G is a 2-connected graph, then c ¸ minfn; 2±g. The bound 2± in (i) was enlarged independently by Bondy (1971), Bermond (1976) and Linial (1976) in terms of ¾2 - the minimum degree sum of two nonadjacent vertices: (ii) if G is a 2-connected graph, then c ¸ minfn; ¾2g. In this paper two further extensions of (i) and (ii) are presented by incorporating p and the length of a vine on a longest path of G as new parameters along with n, ± and ¾2.
Keywords: Mathematics; Graph theory