A general sufficient condition for a graph G with λ _m (G) ⩽ ζ _m (G)

Abstract

It has been shown that a λ _m -connected graph G has the property λ _m (G) ⩽ ζ _m (G) for m ⩽ 3. But for m ⩾ 4, Bonsma et al. pointed out that in general the inequality λ _m (G) ⩽ ζ _m (G) is no longer true. Recently Ou showed that any λ ₄-connected graph G with order at least 11 has the property λ ₄(G) ⩽ ζ ₄(G). In this paper, by investigating some structure properties of a λ _m -connected graph G with λ _m (G) > ζ _m (G), we obtain easily the above result, furthermore, we show that every λ _m -connected graph G with order greater than m(m − 1) satisfies the inequality λ _m (G) ⩽ ζ _m (G) for m ⩾ 5. And by constructing some examples, we illustrate that our conditions are the best possible.