An Infinite Family of Sum-Paint Critical Graphs

Abstract

Independently, Zhu and Schauz introduced online list coloring in 2009. In each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm (Painter) gives a new color to an independent subset of these vertices. A graph G is f-paintable for a function $f:V(G)\rightarrow \mathbb {N}$ if Painter can produce a proper coloring when the number of times each vertex v is allowed to be marked is f(v). In 2002, Isaak introduced sum list coloring and the resulting parameter called sum-choosability. The analogous parameter sum-paintability, denoted $\text{s}{\mathring{\text{c}}}\text{h}$, is the minimum of $\sum f(v)$ over all functions f such that G is f-paintable. Always $\text{s}{\mathring{\text{c}}}\text{h}(G)\le |V(G)|+|E(G)|$, and we say that G is sp-greedy when equality holds. When a graph fails to be sp-greedy, any graph containing it as an induced subgraph also fails to be sp-greedy. A graph is sp-critical when it is not sp-greedy but all of its proper induced subgraphs are sp-greedy. We prove the existence of an infinite family of sp-critical graphs. As a corollary, we prove that neither sp-greedy, nor sc-greedy, graphs can be characterized by forbidding a finite family of induced subgraphs.