Discussiones Mathematicae Graph Theory 32(4) (2012) 659-676
doi: 10.7151/dmgt.1637

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Structural Results on Maximal k-Degenerate Graphs

Allan Bickle

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


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

2010 Mathematics Subject Classification: 05C75.


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Received 9 June 2011
Revised 13 December 2011
Accepted 13 December 2011