AN EXACT LOWER-BOUND ON THE NUMBER OF CUT-SETS IN MULTIGRAPHS

A cut-set in an undirected multigraph G is a subset of edges whose removal makes the graph disconnected. Let m(i)(G) denote the number of all cut-sets, each of which consists of i edges. In this paper, for any multigraph G with n nodes and e (greater-than-or-equal-to 3n/2) edges, we show that m(i)(G) with alpha less-than-or-equal-to i less-than-or-equal-to 2(alpha - gamma) - 3, where alpha = left perpendicular 2e/n right perpendicular and gamma = left perpendicular 2e/(n(n - 1)) right perpendicular, is greater than or equal to [GRAPHICS] ((alpha + 1)n - 2e) + [GRAPHICS] (2e - alphan). A necessary and sufficient condition for a multigraph G with given n and e to minimize m(i)(G) for an i with i less-than-or-equal-to 2(alpha - gamma) - 3 is presented. We also show that there exists a graph such that the lower bound is tight for all i in the range [alpha, 2(alpha - gamma) - 3]. (C) 1994 John Wiley & Sons, Inc.