Chromatic polynomials of complements of bipartite graphs

Details Chromatic polynomials of complements of bipartite graphs Chromatic polynomials of complements of bipartite graphs

2011-03-07

arxiv.org/abs/1103.1355v4

Mathematics

Combinatorics

15 pages; significantly revised and expanded (with thanks to the referees). To appear in Graphs and Combinatorics

Scientific paper

Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the $\alpha +n$ conjecture, by showing that for any cubic integer $\alpha$, there is a natural number $n$ such that $\alpha +n$ is a chromatic root.

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