Probabilistic Methods for Decomposition Dimension of Graphs

Abstract.

In a graph G, the distance from an edge e to a set F⊆E(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct. For the complete graph K _n and the k-dimensional hypercube Q _k , we prove that (2−o(1))lgn≤dec(K _n )≤(3.2+o(1))lgn and k/lgk≤ dec(Q _k )≤ (3.17+o(1))k/lgk. The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that p n ^1−ɛ→∞ for some positive constant ɛ have decomposition dimension Θ(lnn) with high probability.