# Structural results on maximal k-degenerate graphs

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 4, page 659-676
- ISSN: 2083-5892

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topAllan Bickle. "Structural results on maximal k-degenerate graphs." Discussiones Mathematicae Graph Theory 32.4 (2012): 659-676. <http://eudml.org/doc/270959>.

@article{AllanBickle2012,

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.},

author = {Allan Bickle},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {k-degenerate; k-core; k-tree; degree sequence; Ramsey number; -degenerate; -core; -tree},

language = {eng},

number = {4},

pages = {659-676},

title = {Structural results on maximal k-degenerate graphs},

url = {http://eudml.org/doc/270959},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Allan Bickle

TI - Structural results on maximal k-degenerate graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 4

SP - 659

EP - 676

AB - 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.

LA - eng

KW - k-degenerate; k-core; k-tree; degree sequence; Ramsey number; -degenerate; -core; -tree

UR - http://eudml.org/doc/270959

ER -

## References

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