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22nd Cumberland Conference on Combinatorics, Graph Theory and Computing
May 21-23, 2009
Western Kentucky University
Bowling Green, KY, USA

Organizers
Bela Csaba, Chair; Mustafa Atici; Robert Crawford; Claus Ernst; Dominic Lanphier; Attila Por

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Defining Parameters for Countably Infinite Graphs
by
Peter J. Slater
University of Alabama in Huntsville

Because one can tile the infinite square grid ZxZ that is regular of degree four with stars K(1, 4), it seems reasonable to define the domination percentage parameter of ZxZ to be gamma*(ZxZ) = 1/5. Using a different tiling of ZxZ one can argue that the locating- dominating percentage value is LD*(ZxZ) = 3/10. It also seems obvious that for independence we have BETA*(ZxZ) = 1/2.

However, as will be discussed here, it is less than obvious how to define percentage parameters for countably infinite graphs, even when they are locally finite, are of bounded degree, or even are regular of degree three.

Date received: April 14, 2009

Copyright © 2009 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cayq-20.