Sum choice number of generalized θ-graphs

Abstract Let G = ( V E ) be a simple graph and for every vertex v ∈ V let L ( v ) be a set (list) of available colors. G is called L -colorable if there is a proper coloring φ of the vertices with φ ( v ) ∈ L ( v ) for all v ∈ V . A function f : V → N is called a choice function of G and G is said to be f -list colorable if G is L -colorable for every list assignment L choice function is defined by size ( f ) = ∑ v ∈ V f ( v ) and the sum choice number χ s c ( G ) denotes the minimum size of a choice function of G . Sum list colorings were introduced by Isaak in 2002 and got a lot of attention since then. For r ≥ 3 a generalized θ k 1 k 2 … k r -graph is a simple graph consisting of two vertices v 1 and v 2 connected by r internally vertex disjoint paths of lengths k 1 , k 2 , … , k r ( k 1 ≤ k 2 ≤ ⋯ ≤ k r ) . In 2014, Carraher et al. determined the sum-paintability of all generalized θ -graphs which is an online-version of the sum choice number and consequently an upper bound for it. In this paper we obtain sharp upper bounds for the sum choice number of all generalized θ -graphs with k 1 ≥ 2 and characterize all generalized θ -graphs G which attain the trivial upper bound | V ( G ) | + | E ( G ) | .