Sum-Paintability of Generalized Theta-Graphs

In online list coloring [introduced by Zhu (Electron J Comb 16(1):#R127, 2009) and Schauz (Electron J Comb 16:#R77, 2009)], on each round the set of vertices having a particular color in their lists is revealed, and the coloring algorithm chooses an independent subset of this set to receive that color. For a graph and a function , the graph is -paintable if there is an algorithm to produce a proper coloring when each vertex is allowed to be presented at most times. The sum-paintability of , denoted , is is -paintable. Basic results include for every graph and when are the blocks of . Also, adding an ear of length to adds to the sum-paintability, when . Strengthening a result of Berliner et al., we prove . The generalized theta-graph consists of two vertices joined by internally disjoint paths of lengths . A book is a graph of the form , denoted when there are internally disjoint paths of length 2. We prove . We use these results to determine the sum-paintability for all generalized theta-graphs.