Discussiones Mathematicae Graph Theory 33(1) (2013) 147-165
doi: 10.7151/dmgt.1674

Dedicated to Mietek Borowiecki on the occasion of his seventieth birthday.

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On maximum weight of a bipartite graph
of given order and size

Mirko Horňák
Stanislav Jendrol'

Institute of Mathematics
P.J. Šafárik University
Jesenná 5, 04001 Košice, Slovakia

Ingo Schiermeyer

Institut für Diskrete Mathematik und Algebra
Technische Universität Bergakademie Freiberg
09596 Freiberg, Germany

Abstract

The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erdos was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m)+1.

Keywords: weight of an edge, weight of a graph, bipartite graph

2010 Mathematics Subject Classification: 05C35.

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Received 2 February 2012
Revised 8 January 2012
Accepted 9 January 2012