Sum-Paintability of Generalized Theta-Graphs

Abstract

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 $G$ and a function $f:\,V(G)\rightarrow {\mathbb N}$, 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 $\chi _{sp}(G)$, is $\min \{\sum f(v):\,G$ is $f$-paintable$\}$. Basic results include $\chi _{sp}(G)\le |V(G)|+|E(G)|$ for every graph $G$ and $\chi _{sp}(G)=(\sum _{i=1}^{k} \chi _{sp}(H_i))-(k-1)$ when $H_{1},\ldots ,H_{k}$ are the blocks of $G$. Also, adding an ear of length $\ell$ to $G$ adds $2\ell -1$ to the sum-paintability, when $\ell \ge 3$. Strengthening a result of Berliner et al., we prove $\chi _{sp}(K_{2,r})=2r + \min \{l+m:\,lm>r\}$ . The generalized theta-graph $\Theta _{\ell _{1},\ldots ,\ell _{k}}$ consists of two vertices joined by internally disjoint paths of lengths $\ell _{1},\ldots ,\ell _{k}$. A book is a graph of the form $\Theta _{1,2,\dots ,2}$, denoted $B_r$ when there are $r$ internally disjoint paths of length 2. We prove $\chi _{sp}(B_r)=2r + \min _{l,m\in \mathbb {N}}\{l+m:\,m(l-m)+{m\atopwithdelims ()2}>r\}$. We use these results to determine the sum-paintability for all generalized theta-graphs.