On -hued coloring of -minor free graphs

A list assignment L of G is a mapping that assigns every vertex v ∈ V ( G ) a set L ( v ) of positive integers. For a given list assignment L of G , an ( L , r )-coloring of G is a proper coloring c such that for any vertex v with degree d ( v ) , c ( v ) ∈ L ( v ) and v is adjacent to at least min { d ( v ) , r } different colors. The r -hued chromatic number of G , χ r ( G ) , is the least integer k such that for any v ∈ V ( G ) with L ( v ) = { 1 , 2 , … , k } , G has an ( L , r ) -coloring. The r -hued list chromatic number of G , χ L , r ( G ) , is the least integer k such that for any v ∈ 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 , and K ( r ) = ⌊ 3 r / 2 ⌋ + 1 if r ≥ 4 . We proved that if G is a K 4 -minor free graph, then(i) G ) ≤ K ( r ) , and the bound can be attained; r ( G ) ≤ K ( r ) + 1 . extends a former result in Lih et al. (2003).