Perfect graphs of fixed density: counting and homogenous sets
Abstract
For c in [0,1] let P_n(c) denote the set of nvertex perfect graphs with density c and C_n(c) the set of nvertex graphs without induced C_5 and with density c. We show that logP_n(c)/binom{n}{2}=logC_n(c)/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(2c1)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomassé [Almost all Hfree graphs have the ErdősHajnal property] in the case of forbidden induced C_5.
 Publication:

arXiv eprints
 Pub Date:
 February 2011
 arXiv:
 arXiv:1102.5229
 Bibcode:
 2011arXiv1102.5229B
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 19 pages