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ArticleOriginal scientific text

Title

Structural results on maximal k-degenerate graphs

Authors 1

Affiliations

  1. Department of Mathematics, Western Michigan University, 1903 W. Michigan, Kalamazoo, MI 49008

Abstract

A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. These graphs were introduced by Lick and White in 1970 and have been studied in several subsequent papers. We present sharp bounds on the diameter of maximal k-degenerate graphs and characterize the extremal graphs for the upper bound. We present a simple characterization of the degree sequences of these graphs and consider related results. Considering edge coloring, we conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and prove this in some special cases. We present some results on decompositions and arboricity of maximal k-degenerate graphs and provide two characterizations of the subclass of k-trees as maximal k-degenerate graphs. Finally, we define and prove a formula for the Ramsey core numbers.

Keywords

ENG
k-degenerate, k-core, k-tree, degree sequence, Ramsey number

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Discussiones Mathematicae Graph Theory
2012/32/4
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Pages:
659-676
Main language of publication
English
Received
2011-06-09
Accepted
2011-12-13
Published
2012

DOI: 10.7151/dmgt.1637

Exact and natural sciences