Sum choice number of generalized [formula omitted]-graphs

Abstract

Let G=(VE) 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∈Vf(v) and the sum choice numberχsc(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 generalizedθk1k2…kr-graph is a simple graph consisting of two vertices v1 and v2 connected by r internally vertex disjoint paths of lengths k1,k2,…,kr(k1≤k2≤⋯≤kr).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 k1≥2 and characterize all generalized θ-graphs G which attain the trivial upper bound |V(G)|+|E(G)|.