On r-hued list coloring of K4(7)-minor free graphs

For a given list assignment L of a graph G, an (L, r)-coloring of G is a proper coloring c such that for any vertex v with degree d(v), v is adjacent to vertices of at least min{d(v), r} different color with c(v) is an element of L(v). The r-hued list chromatic number of G, denoted as chi(L,r)(G), is the least integer k, such that for any v is an element of V (G) and every list assignment L with |L(v)| = k, G has an (L, r)-coloring. Let K(r) = r + 3 if 2 <= r <= 3, K(r) = (sic)3r/2(sic) + 1 if r >= 4. In Song et al. (2014), it is proved that if G is a K4-minor-free graph, then chi L,r(G) <= K(r) + 1. Let K4(n) be the set of all subdivisions of K4 on n vertices. Utilizing the decompositions by Chen et al for K4(7)-minor free graphs in Chen et al. (2020), we prove that if G is a K4(7)-minor free graph, then chi L,r(G) <= K(r) + 1. (c) 2021 Elsevier B.V. All rights reserved.