A Property on Edge-disjoint Spanning Trees

The graphs in this note are finite and undirected . We allow multiple edges , but forbid loops . We shall use the notation of Bondy and Murty [1] , unless otherwise stated . Let G be a graph with E ( G ) ? [ . Let τ ( G ) denote the number of edge-disjoint spanning trees of G . For X ‘ E ( G ) , the notation G ( X ) denotes the spanning subgraph of G with edge set X , whereas G [ X ] denotes the subgraph of G induced by X . The contraction G / X is the graph obtained from G by identifying the ends of each edge in X and then deleting the resulting loops . When H is a connected subgraph of G , we use G / H for G / E ( H ) . For convenience , we define G / [ 5 G . as in [1] , v ( G ) denotes the number of components of G . By H ‘ G we mean that H is a subgraph of G . Throughout this note , N denotes the set of all positive integers . For a set S , an m partition k X 1 , X 2 , . . . , X m l of S is a collection of m subsets of S such that :