Biclosure and bistability in a balanced bipartite graph

Abstract

Abstract The k ‐biclosure of a balanced bipartite graph wiht color classes A and B is the graph obtained from G by recursively joining pairs of nonadjacent vertices respectively taken in A and B whose degree sum is at least k , until no such pair remains. A property P defined on all the balanced bipartite graphs of order 2 n is k ‐bistable if whenever G + ab has property P and d G ( b ) ≧ k then G itself has property P . We present a synthesis of results involving, for some properties, P , the bistability of P , the k ‐biclosure of G , the number of edges and the minimum degree. © 1995 John Wiley & Sons, Inc.