Probabilistic Methods for Decomposition Dimension of Graphs

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 Kn and the k-dimensional hypercube Qk, we prove (2− o(1)) lgn ≤ dec(Kn) ≤ (3.2 + o(1)) lgn and k/ lg k ≤ dec(Qk) ≤ (3.17 + o(1))k/ lg k. The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that pn1−ε →∞ for some positive constant ε have decomposition dimension Θ(lnn) with high probability. AMS classifications: 05C12, 05C35, 05D05, 05D40