Probabilistic Methods for Decomposition Dimension of Graphs

Abstract

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 (2The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that pn 1−ε → ∞ for some positive constant ε have decomposition dimension Θ(ln n) with high probability.

Decomposition dimension of corona product of some classes of graphs

“…M. Hagita, A. Kundgen and D. B. West [3] used probabilistic methods to obtain upper bounds for decomposition dimension of complete graphs and regular graphs. H. Enomoto and T. Nakamigawa [2] established a lower bound for decomposition dimension of graphs using the maximum degree of G. They proved that for any graph G, dec(G) ≥ dlog 2 ∆(G)e + 1.…”