Fractional acquisition in graphs

Abstract Let G be a vertex-weighted graph in which each vertex has weight 1 . Given a vertex u with positive weight and a neighbor v whose weight is at least the weight on u , a fractional acquisition move transfers some amount of weight at u from u to v . The fractional acquisition number of G , written a f ( G ) , is the minimum number of vertices with positive weight after a sequence of fractional acquisition moves in G . In this paper, we determine the fractional acquisition number of all graphs: if G is an n -vertex path or cycle, then a f ( G ) = ⌈ n / 4 ⌉ ; if G is connected with maximum degree at least 3 , then a f ( G ) = 1 .