Bounding the list color function threshold from above

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The chromatic polynomial of a graph G, denoted P (G, m), is equal to the number of proper m-colorings of G for each m ∈ N. In 1990, Kostochka and Sidorenko introduced the list color function of graph G, denoted P ℓ (G, m), which is a list analogue of the chromatic polynomial. The list color function threshold of G, denoted τ (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, τ (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 τ . In this paper we develop tools for bounding the list color function threshold of complete bipartite graphs from above. We show that for any n ≥ 2, τ (K 2,n ) ≤ ⌈(n + 2.05)/1.24⌉. Interestingly, our proof makes use of classical results such as Rolle's Theorem and Descartes' Rule of Signs.