A generalization of the Birthday problem and the chromatic polynomial

Details A generalization of the Birthday problem and the chromatic polynomial A generalization of the Birthday problem and the chromatic polynomial

2011-05-03

arxiv.org/abs/1105.0698v2

Mathematics

Combinatorics

23 pages, 9 figures

Scientific paper

The birthday problem states that there is at least half a chance that some two out of twenty-three randomly chosen people will share the same birth date. The calculation for this problem assumes that all birth dates are equally likely. We consider the following two modifications of this question. What if the distribution of birth dates is non-uniform and possibly even unknown? Further, what if we focus on birthdays shared by two friends rather than any two people? In this paper we present some of our results and conjectures in this generalized setting. We will also show how these results are related to the Stanley-Stembridge poset chain chromatic conjecture and the `shameful conjecture', two famous conjectures in combinatorics.

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