SPANNING CLOSED TRAILS IN GRAPHS

Let G be a 2-edge-connected simple graph on n > 95 vertices. Let l be the number of vertices of degree 2 in G. We prove that if l < n/5 - 19 and if, for every edge uv is-an-element-of E(G), d(u) + d(v) greater- than-or-equal-to 2n/5 - 2, then exactly one of the following holds: (a ) G has a spanning closed trail; (b) G can be contracted to K2,c - 2, where c less-than-or-equal-to max {5, 3 + l} is an odd number. An exam ple shows that if a graph satisfies the conditions above except that i t has too many vertices of degree 2, then the conclusion fails. This r esult is related to a conjecture of Benhocine et al. (1986), recently proved by Veldman. We obtain some other related results.