Bounding the list color function threshold from above

The chromatic polynomial of a graph G, denoted by P(G, m), is equal to the number of proper m-colorings of G for each m is an element of N. In 1990, Kostochka and Sidorenko introduced the list color function of graph G, denoted by P-l(G, m), which is a list analogue of the chromatic polynomial. The list color function threshold of G, denoted by tau (G), is the smallest k such that P(G, k) >= 0 and P-l(G, m) = P(G, m) whenever m >= k. It is known that for every graph G, tau(G) is finite, and a recent paper of Kaul et al. suggests that complete bipartite graphs may be the key to understanding the extremal behavior of tau. We develop tools for bounding the list color function threshold of complete bipartite graphs from above. We show that, for any n >= 2, tau (K-2,K-n) <= [(n+2.05)/1.24]. Interestingly, our proof makes use of classical results such as Rolle's theorem and Descartes' rule of signs.