Families of Online Sum-Choice-Greedy Graphs

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 $$G$$G is said to be $$f$$f-paintable for a function $$f:V(G)\rightarrow \mathbb {N}$$f:V(G)→N if there is an algorithm to produce a successful coloring whenever each vertex $$v$$v is allowed to be marked at most $$f(v)$$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 $$\sum f(v)$$∑f(v) over all functions $$f$$f such that $$G$$G is $$f$$f-paintable; we denote this value by $$\chi _{sp}(G)$$χsp(G). Always $$\chi _{sp}(G)\le |V(G)|+|E(G)|$$χsp(G)≤|V(G)|+|E(G)|, and we say that $$G$$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.