Random interval graphs

✍ Scribed by Nicholas Pippenger

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 209 KB
• Volume: 12
• Category: Article
• ISSN: 1042-9832
• DOI: 10.1002/(sici)1098-2418(199807)12:4<361::aid-rsa4>3.0.co;2-r

✦ Synopsis

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy Ž period, while the size K of the largest clique which for interval graphs is equal to the . chromatic number corresponds to the maximum number of customers in the system during a busy period. We obtain the following results for both the MrDrϱ and the MrMrϱ models, with arrival rate per mean service time. The expected number of vertices is e , and the distribution of the Nre converges pointwise to an exponential distribution with mean 1 as tends to infinity. This implies that the distribution of log N y converges Ž .

2 pointwise to a distribution with mean y␥ where ␥ is Euler's constant and variance r6. w x The size K of the largest clique falls in the interval e y 2 log , e q 1 with probability tending to 1 as tends to infinity. Thus the distribution of the ratio Krlog N converges pointwise to that of the constant e, in contrast to the situation for random graphs generated by unbiased coin flips, in which the distribution of Krlog N converges pointwise to that of the constant 2rlog 2.