Total list weighting of graphs with bounded maximum average degree

Abstract A proper total weighting of a graph G is a mapping ϕ which assigns to each vertex and each edge of G a real number as its weight so that for any edge u v of G , ∑ e ∈ E ( v ) ϕ ( e ) + ϕ ( v ) ≠ ∑ e ∈ E ( u ) ϕ ( e ) + ϕ ( u ) . A ( k , k ′ ) -list assignment of G is a mapping L which assigns to each vertex v a set L ( v ) of k permissible weights and to each edge e a set L ( e ) of k ′ permissible weights. An L -total weighting is a total weighting ϕ with ϕ ( z ) ∈ L ( z ) for each z ∈ V ( G ) ∪ E ( G ) . A graph G is called ( k , k ′ ) -choosable if for every ( k , k ′ ) -list assignment L of G , there exists a proper L -total weighting. It was proved in Tang and Zhu (2017) that if p ∈ { 5 , 7 , 11 } , a graph G without isolated edges and with mad ( G ) ≤ p − 1 is ( 1 , p ) -choosable. In this paper, we strengthen this result by showing that for any prime p , a graph G without isolated edges and with mad ( G ) ≤ p − 1 is ( 1 , p ) -choosable.