Closure and Spanning Trees with Bounded Total Excess

Let α≥ 0 and k ≥ 2 be integers. For a graph G , the total k -excess of G is defined as te (G;k)=∑ _v ∈ V(G)max{d_G(v)-k,0} . In this paper, we propose a new closure concept for a spanning tree with bounded total k -excess. We prove that: Let G be a connected graph, and let u and v be two non-adjacent vertices of G . If G satisfies one of the following conditions, then G has a spanning tree T such that te (T;k) ≤α if and only if G+uv has a spanning tree T' such that te (T';k) ≤α : max{∑ _x ∈ X d_G(x): X is a subset of S with |X|=k }≥ |G|-1 for every independent set S in G of order k+1 such that { u,v }⊆ S ; or max{∑ _x ∈ X d_G(x): X is a subset of S with |X|=k }≥ |G|-α -1 for every independent set S in G of order k+α +1 such that S ∩{ u,v }∅ . We also show examples to show that these conditions are sharp.