University of Illinois Urbana-Champaign

Extremal Results and Algorithms for Degree Sequences of Graphs

Will, Todd Gerald

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Description

Title
Extremal Results and Algorithms for Degree Sequences of Graphs
Author(s)
Will, Todd Gerald
Issue Date
1993
Doctoral Committee Chair(s)
West, Douglas B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2014-12-17T23:17:48Z
Keyword(s)
Mathematics
Abstract
  • A 2-multigraph is a loopless multigraph with maximum multiplicity 2; pairs of vertices induce 0, 1, or 2 edges. A 2-multigraph is parsimonious if it has the minimum number of single edges (multiplicity 1) among all 2-multigraphs with the same degree sequence. In every parsimonious 2-multigraph, the subgraph of single edges consists of isolated stars and possibly one component that is a triangle. We prove the conjecture of Brualdi and Michael that for any fixed degree sequence, either every parsimonious 2-multigraph with those vertex degrees has a triangle of single edges, or no such parsimonious 2-multigraph has a triangle of single edges.
  • We determine for a planar graph on n vertices the maximum values for the following: (1) The sum of the m largest vertex degrees. (2) For k $\ge$ 12, the number of vertices of degree at least k and the sum of the degrees of vertices with degree at least k. In addition, for 6 $\le$ k $\le$ 11, we determine upper and lower bounds for the latter two values, which match for certain congruence classes of n.
  • Berge showed that if two graphs G and H have the same degree sequence, then G can be transformed into H by a sequence of elementary degree-preserving transformations. We show that computing the minimum length of such a sequence is an NP-complete problem. In addition we disprove a conjecture of John Gimbel on an analogous result for oriented graphs, and obtain partial results toward a revised conjecture.
Graduation Semester
1993
Type of Resource
text
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http://hdl.handle.net/2142/72546

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