Lower bounds on the independence number of certain graphs of odd girth at least seven

Abstract

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233237] proved that every connected subcubic triangle-free graph G has an independent set of order at least (4n(G)-m(G)-1)7 where n(G) and m(G) denote the order and size of G, respectively. We conjecture that every connected subcubic graph G of odd girth at least seven has an independent set of order at least (5n(G)-m(G)-1)9 and verify our conjecture under some additional technical assumptions. © 2010 Elsevier B.V. All rights reserved.