A degree sum condition for longest cycles in 3‐connected graphs

Abstract

Abstract:For a graph G, we denote by d G (x) and κ(G) the degree of a vertex x in G and the connectivity of G, respectively. In this article, we show that if G is a 3-connected graph of order n such thatfor every independent set {x, y, z}, then G contains a cycle of length at least min{d − κ(G), n}.

Forbidden pairs and the existence of a dominating cycle

“…On the other hand, one often try to find a dominating cycle in order to find a Hamilton cycle in a given graph (recall that a cycle C in a graph G is dominating if every edge of G is incident with a vertex of C). For example, if some longest cycle in a graph G is dominating and the independence number of G is at most its minimum degree, then G has a Hamilton cycle (the related results can be found in [4,24]). It is also shown that the dominating cycle conjecture that "every cyclically 4-edge-connected cubic graph has a dominating cycle" by Fleischner [13] is equivalent to not only the well-known conjecture that "every 4-connected K 1,3 -free graph is…”