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

Heckman and Thomas [C.C. Heckman, R. Thomas, A new proof of the independence ratio of triangle-free cubic graphs, Discrete Math. 233 (2001) 233–237] proved that every connected subcubic triangle-free graph G has an independent set of order at least ( 4 n ( 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 ( 5 n ( G ) − m ( G ) − 1 ) / 9 and verify our conjecture under some additional technical assumptions.