Long Dominating Cycles in Graphs

Let G be a connected graph of order n, and NC2(G) denote min{|N(u)∪N(v)| : dist(u,v) = 2}, where dist(u,v) is the distance between u and v in G. A cycle C in G is called a dominating cycle, if V (G)\V (C) is an independent set in G. In this paper, we prove that if G contains a dominating cycle and � ≥ 2, then G contains a dominating cycle of length at least min{n,2NC2(G) −1} and give a family of graphs showing our result is sharp, which proves a conjecture of R. Shen and F. Tian, also related with the cyclic structures of algebraically Smarandache multi-spaces. subscript G of NG(H). We denote by G(S) the subgraph of G induced by any subset S of V (G). For a connected graph G and u, v ∈ V (G), we define the distance between u and v in G, denoted by dist(u, v), as the minimum value of the lengths of all paths joining u and v in G. If G is non-complete, let NC(G) denote min{|N(u, v)| : uv / ∈ E(G)} and NC2(G) denote min{|N(u, v)| : dist(u, v) = 2}; if G is complete, we set NC(G) = n −1 and NC2(G) = n −1. In (2), Broersma and Veldman gave the following result.