Families of Online Sum-Choice-Greedy Graphs

Abstract

In online list coloring (introduced by Zhu and by Schauz in 2009), at each step a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of the marked vertices to receive that color. A graph $G$ is said to be $f$-paintable for a function $f:V(G)\rightarrow \mathbb {N}$ if there is an algorithm to produce a successful coloring whenever each vertex $v$ is allowed to be marked at most $f(v)$ times. In 2002 Isaak introduced sum list coloring and the resulting parameter called sum-choosability. The analogous notion of online sum-choosability, or sum-paintability, is the minimum of $\sum f(v)$ over all functions $f$ such that $G$ is $f$-paintable; we denote this value by $\chi _{sp}(G)$. Always $\chi _{sp}(G)\le |V(G)|+|E(G)|$, and we say that $G$ is sp-greedy when equality holds. We conjecture that all outerplanar graphs are sp-greedy. We prove this for every outerplanar graph whose weak dual is a path and give further restrictions on the structure of a minimal counterexample. We also prove that wheels are sp-greedy.