Chvatal-Erdos Conditions and Almost Spanning Trails

Let alpha '(G),ess '(G),kappa(G),kappa '(G),NG(v) and Di(G) denote the matching number, essential edge connectivity, connectivity, edge connectivity, the set of neighbors of v in G and the set of degree i vertices of a graph G, respectively. For u,v is an element of V(G), define u similar to v if and only if u=v or both u,v is an element of D2(G) and NG(u)=NG(v). Then, similar to is an equivalence relation, and [v] denotes the equivalence class containing v. A subgraph H of G is almost spanning if H subset of G-D1(G),& Union;j >= 3Dj(G)subset of V(H) and for any v is an element of D2(G),|[v]-V(H)|<= 1. The line graph version of Chvatal-Erdos theorem for a connected graph G are extended as follows. If ess '(G)>=alpha '(G), then has an almost spanning closed trail. If ess '(G)>=alpha '(G)-1, then has an almost spanning trail. If ess '(G)>=alpha '(G)+1, then for e,e 'is an element of E(G-D1(G)),G-D1(G) has an almost spanning trail starting from and ending at e '.