Characterizations of arboricity of graphs

The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of trees or forests dates back to Nash-Williams and Tutte [6, 7, 11]. Since then, many authors have examined various tree decompositions of classes of graphs (for example [2, 8]). The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Graphs in this paper will include those with multiple edges but no loops. Let VG and EG be respectively, the number of vertices and edges in the graph G. In [1], Albertson and Haas define a graph G to be bounded by the function f(n) if EG = f(VG) and each subgraph H ⊂ G satisfies EH ≤ f(VH). That paper begins the study of which functions bound graphs, and which bounding functions correspond to properties of graphs. In [3], Catlin et al. characterize uniformly dense graphs by a bounding function. This paper characterizes graphs bounded by functions of the form k(VG−1)− l for integers k ≥ l ≥ 0. There are many cases in which the condition that G is bounded by a function