Obtaining Uniformly Dense Graphs

Let G be a non-trivial, loop-less multi-graph and for each non-trivial sub-graph H of G, let g(H) = |E(H)| |V (H)|−ω(G) . G is said to be uniformly dense if and only if γ(G), the maximum among g(H) taken over all non-trivial subgraphs H of G is attained when H = G. This quantity γ(G) is called the fractional arboricity of G and was introduced by Catlin, Grossman, Hobbs and Lai [4]. γ(G)− g(G) measures how much the given graph G is away from being uniformly dense. In this paper, we describe a systematic method of modifying a given graph to obtain a uniformly dense graph on the same number of vertices and edges. We obtain this by a sequence of steps; each step re-defining one end-vertex of an edge in the given graph. After each step, either the value γ of the new graph formed is lesser than that of the graph from the previous step or the size of the maximal γ-achieving subgraph of the new graph is smaller than that of the graph in the previous step. We will see that at most O(|V (G)|) steps results in a uniformly dense graph.