Weighted Graphs Permitting No LEW-Embeddings

A weighted graph embedded in a surface is called LEW-embedded if every noncontractible cycle is longer than every facial walk. C.Thomassen studied LEW-embeddings of unweighted graphs. But little is known for those having nonconstant weight function. In this article we study the LEW-embeddability of weighted grads graph G(a, b)(a≤2, b≥2) and the Mobius ladder graph Gn(n≥4) and show that such two types of weighted graphs have no LEW-embeddings. Based on these two kinds of graphs we construct weighted graphs which are strongly embedded in Sn and Nn and permit no LEW-embeddings in the same surface they embedded. Finally we study a kind of cycle base of complete graph Kn and prove that all the directed cycles of strongly connected graph D generate cycle space of the corresponding base graph G.