The list r-hued coloring of Km,n

Let L be list assignment of colors available for vertices of a graph G. An (L, r) -coloring of G is a proper coloring c such that for any vertex v is an element of V (G), we have c(v) is an element of L(v) and | c(NG(v)) |>= min{dG(v), r}. The list r -hued chromatic number of G, denoted as chi L,r(G), is the least integer k, such that for list assignment L satisfying | L(v) |= k, for any v is an element of V (G), G has an (L, r) -coloring. Let Km,n denote a complete bipartite graph. In [Discrete Math. 306(16)(2006) 1997-2004], it has been proved if n >= m >= 2, then chi r(Km,n)=min{2r, n + m, r + m}. In this paper, we determine the list r -hued chromatic number of Km,n. (c) 2024 Elsevier B.V. All rights reserved.