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On the regulation number of a multigraph

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Abstract

The regulation number of a multigraphG having maximum degreed is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG. It is shown that the regulation number of any multigraph is at most 3. The regulation number of a multidigraph is defined analogously and is shown never to exceed 2. A multigraphG has strengthm if every two distinct vertices ofG are joined by at mostm parallel edges. For a multigraphG of strengthm and maximum degreed, them-regulation number ofG is the minimum number of additional vertices that are required to construct ad-regular supermultigraph ofG having strengthm. A sharp upper bound on the 2-regulation number of a multigraph is shown to be (d+5)/2, and a conjecture for generalm is presented.

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Authors and Affiliations

  1. Department of Mathematics, Western Michigan University, 49008, Kalamazoo, MI, USA

    Gary Chartrand & Ortrud R. Oellermann

  2. Department of Mathematics, The University of Michigan, 48109, Ann Arbor, MI, USA

    Frank Harary

Authors
  1. Gary Chartrand
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  2. Frank Harary
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  3. Ortrud R. Oellermann
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Additional information

Research supported by a Western Michigan University faculty research fellowship.

Research Professor of Electrical Engineering and Computer Science, Stevens Institute, Hoboken, NJ and Visiting Scholar, Courant Institute, New York University, Spring 1984.

Research supported in part by a Western Michigan University research assistantship from the Graduate College and the College of Arts and Sciences.

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Chartrand, G., Harary, F. & Oellermann, O.R. On the regulation number of a multigraph. Graphs and Combinatorics 1, 137–144 (1985). https://doi.org/10.1007/BF02582938

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  • DOI: https://doi.org/10.1007/BF02582938

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