Rubbling and Optimal Rubbling of Dense Bipartite Graphs

Abstract

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices u and w that are adjacent to a vertex v, and an extra pebble is added at vertex v. The rubbling number of G, denoted by ρ(G), is the smallest number m such that for every distribution of m pebbles on G and every vertex v, at least one pebble can be moved to v by a sequence of rubbling moves. The optimal rubbling number of G, denoted by ρ_opt(G), is the smallest number k such that for some distribution of k pebbles on G, one pebble can be moved to any vertex of G. In this paper, we determine ρ(G) for a non-complete bipartite graph G ∈ B(s, t) with \(\delta(G)\geq\lceil\frac{2s+1}{3}\rceil\), give an upper bound of ρ(G) for G ∈ B(s, t) with \(\delta(G)\geq\lceil\frac{s+1}{2}\rceil\), and also obtain ρ_opt(G) for a non-complete bipartite graph G ∈ B(s, t) with \(\delta(G)\geq\lceil\frac{s+1}{2}\rceil\), where B(s, t) is the set of all connected bipartite graphs with partite sets of size s and t (s ≥ t) and δ(G) is the minimum degree of G.