On dense strongly Z2s-1-connected graphs

Let G be a graph and s > 0 be an integer. If, for any function b : V (G) → Z2s+1 satisfying  v∈V (G) b(v) ≡ 0 (mod 2s+1), G always has an orientation D such that the net outdegree at every vertex v is congruent to b(v)mod 2s+1, then G is strongly Z2s+1-connected. For a graph G, denote by α(G) the cardinality of a maximum independent set of G. In this paper, we prove that for any integers s, t > 0 and real numbers a, bwith 0 < a < 1, there exist an integer N(a, b, s) and a finite family Y(a, b, s, t) of non-strongly Z2s+1-connected graphs such that for any connected simple graph G with order n ≥ N(a, b, s) and α(G) ≤ t , if G satisfies one of the following conditions: (i) for any edge uv ∈ E(G), max{dG(u), dG(v)} ≥ an + b, or (ii) for any u, v ∈ V (G) with distG(u, v) = 2, max{dG(u), dG(v)} ≥ an + b, then G is strongly Z2s+1-connected if and only if G is not contractible to a member in the finite family Y(a, b, s, t). © 2015 Elsevier B.V. All rights reserved.