Sum-Paintability of Generalized Theta-Graphs

In online list coloring (introduced by Zhu and by Schauz in 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 G and a function f : V (G) → N, the graph is f -paintable if there is an algorithm to produce a proper coloring when each vertex v is allowed to be presented at most f(v) times. The sum-paintability of G, denoted χsp(G), is min{ ∑ f(v) : G is f -paintable}. Basic results include χsp(G) ≤ |V (G)| + |E(G)| for every graph G and χsp(G) = ( ∑k i=1 χsp(Hi))− (k − 1) when H1, . . . ,Hk are the blocks of G. Also, adding an ear of length l to G adds 2l− 1 to the sum-paintability, when l ≥ 3. Strengthening a result of Berliner et al., we prove χsp(K2,r) = 2r+min{l+m : lm > r}. The generalized theta-graph Θl1,...,lk consists of two vertices joined by internally disjoint paths of lengths l1, . . . , lk. A book is a graph of the form Θ1,2,...,2, denoted Br when there are r internally disjoint paths of length 2. We prove χsp(Br) = 2r + minl,m∈N{l + m : m(l − m) + ( m 2 ) > r}. We use these results to determine the sumpaintability for all generalized theta-graphs.