A Tur'an-type problem for circular arc graphs
Abstract
A circular arc graph is the intersection graph of a collection of connected arcs on the circle. We solve a Tur'an-type problem for circular arc graphs: for n arcs, if m and M are the minimum and maximum number of arcs that contain a common point, what is the maximum number of edges the circular arc graph can contain? We establish a sharp bound and produce a maximal construction. For a fixed m, this can be used to show that if the circular arc graph has enough edges, there must be a point that is covered by at least M arcs. In the case m=0, we recover results for interval graphs established by Abbott and Katchalski (1979). We suggest applications to voting situations with interval or circular political spectra.
Keywords
Cite
@article{arxiv.1110.4205,
title = {A Tur'an-type problem for circular arc graphs},
author = {Rosalie Carlson and Stephen Flood and Kevin O'Neill and Francis Edward Su},
journal= {arXiv preprint arXiv:1110.4205},
year = {2011}
}
Comments
18 pages, 8 figures, related papers at http://www.math.hmc.edu/~su/papers.html