The list-coloring function of signed graphs

Abstract

It is known that, for any k-list assignment L of an m-edge graph G, the number of L-list colorings of G is at least the number of the proper k-colorings of G when k>(m−1)/ln⁡(1+2). In this paper, we extend the Whitney's broken cycle theorem to L-colorings of signed graphs, by which we show that if k>(m3)+(m4)+m−1 then, for any k-assignment L, the number of L-colorings of a signed graph Σ with m edges is at least the number of the proper k-colorings of Σ. Further, if L is 0-free (resp., 0-included) and k is even (resp., odd), then the lower bound (m3)+(m4)+m−1 for k can be improved to (m−1)/ln⁡(1+2).