On dense strongly Z(2s+i)-connected graphs

摘要

Let G be a graph and s > 0 be an integer. If, for any function b : V (G) --> Z(2s+1) satisfying Sigma(v is an element of 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 Z(2s+1)-connected. For a graph G, denote by alpha(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, b with 0 < a < 1, there exist an integer N (a, b, s) and a finite family y (a, b, s, t) of non-strongly Z(2s+1)-connected graphs such that for any connected simple graph G with order n >= N (a, b, s) and alpha(G) <= t, if G satisfies one of the following conditions: