Total weight choosability of graphs with bounded maximum average degree

Abstract A total weighting of a graph G is a function ϕ that assigns a weight to each vertex and each edge of G . The vertex-sum of a vertex v with respect to ϕ is S ϕ ( v ) = ϕ ( v ) + ∑ e ∈ E ( v ) ϕ ( e ) , where E ( v ) is the set of edges incident to v . A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph G is ( k , k ′ ) -choosable if the following is true: Whenever each vertex x is assigned a set L ( x ) of k real numbers and each edge e is assigned a set L ( e ) of k ′ real numbers, there is a proper total weighting ϕ of G with ϕ ( y ) ∈ L ( y ) for all y ∈ V ( G ) ∪ E ( G ) . In this paper, we prove that for p ∈ { 5 , 7 , 11 } , a graph G without isolated edges and with mad ( G ) ≤ p − 1 is ( 1 , p ) -choosable. In particular, triangle-free planar graphs are ( 1 , 5 ) -choosable.