The list-coloring function of signed graphs

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 Gwhen k >(m - 1)/ln(1 + root 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 > (m 3) + (m 4) + m - 1 then, for any k-assignment L, the number of L-colorings of a signed graph Sigma with medges is at least the number of the proper k-colorings of Sigma. Further, if Lis 0-free (resp., 0-included) and k is even (resp., odd), then the lower bound (m 3) + (m 4) + m - 1 for kcan be improved to (m - 1)/ ln(1 + root 2). (c) 2023 Published by Elsevier B.V.