Circumferences of k-connected graphs involving independence numbers

Abstract

Let G be a k-connected graph of order n, α: = α(G) the independence number of G, and c(G) the circumference of G. Chvatal and Erdős proved that if α⩽kthen G is hamiltonian. For α⩾k⩾2, Fouquet and Jolivet in 1978 made the conjecture that c(G)⩾k(n+ α − k)/α. Fournier proved that the conjecture is true for α⩽k+ 2 or k = 2 in two different papers. Manoussakis recently proved that the conjecture is true for k = 3. We prove that if G is a k-connected graph, k≥4, of order n and independence number α≥k, then c(G)⩾(k(n+ α − k)/α) − ((k − 3)(k − 4)/2). Consequently, the conjecture of Fouquet and Jolivet holds for k = 4. In addition, we confirm the conjecture for α = k+ 3. Inspired by a result of Kouider, we conjecture that, for every graph G and any two distinct vertices u and v, there is a u − vpath P such that α(G − V(P))⩽α(G) − 1 unless V(G) have a nontrivial partition V1∪V2 satisfying α(G) = α(G[V1]) + α(G[V2]). In this paper, we obtain a partial result regarding this conjecture. Let G be a k-connected graph and k⩾2. While studying the intersection of longest cycles, J. Chen et al. conjectured that, for any two cycles C1 and C2 of G, there are two cycles D1 and D1 such that V(D1)∪V(D2)⊇V(C1)∪V(C2) and |V(D1)∩V(D2)|⩾k. We show that the combination of the above two conjectures implies the conjecture of Fouquet and Jolivet. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:55-76, 2011 © 2011 Wiley Periodicals, Inc.