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On the Maximum Weight of a Sparse Connected Graph of Given Order and Size

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Abstract

The weight of an edge of a graph is defined to be the sum of degrees of vertices incident to the edge. The weight of a graph G is the minimum of weights of edges of G. Jendrol’ and Schiermeyer (Combinatorica 21:351–359, 2001) determined the maximum weight of a graph having n vertices and m edges, thus solving a problem posed in 1990 by Erdős. The present paper is concerned with a modification of the mentioned problem, in which involved graphs are connected and of size \(m\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) -\left( {\begin{array}{c}\lceil n/2\rceil \\ 2\end{array}}\right) \).

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Acknowledgments

This research was supported by the DAAD cooperation contract Freiberg-Košice. The work of the first two authors was supported by Science and Technology Assistance Agency under the contract No. APVV-0023-10 and by Grant VEGA 1/0652/12. The authors are indebted to an anonymous referee for his/her thorough reading of the manuscript and helpful comments.

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Authors and Affiliations

  1. Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 040 01, Košice, Slovakia

    Mirko Horňák & Stanislav Jendrol’

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

    Ingo Schiermeyer

Authors
  1. Mirko Horňák
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  2. Stanislav Jendrol’
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  3. Ingo Schiermeyer
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Correspondence to Mirko Horňák.

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Horňák, M., Jendrol’, S. & Schiermeyer, I. On the Maximum Weight of a Sparse Connected Graph of Given Order and Size. Graphs and Combinatorics 32, 997–1012 (2016). https://doi.org/10.1007/s00373-015-1634-2

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  • DOI: https://doi.org/10.1007/s00373-015-1634-2

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