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.

独立地，Zhu和Schauz在2009年引入了在线列表着色。在每个回合中，标记一组允许接收特定颜色的顶点，并且着色算法（Painter）为这些顶点的独立子集提供了新的颜色。一个图ģ是˚F -上漆一个函数\（F：V（G）\ RIGHTARROW \ mathbb {N} \）如果画家可以产生一个适当的着色时的倍每个顶点的数目v允许被标记是˚F（v）。在2002年，Isaak引入了总和列表着色和产生的参数，称为总和选择性。类似参数求和可涂饰性，表示为\（\文本{S} {\ mathring {\文本{C}}} \文本{H} \） ，是的最小\（\总和F（V）\）以上的所有功能。˚F这样即G ^是˚F -paintable。总是\（\ text {s} {\ mathing {\ text {c}}} \ text {h}（G）\ le | V（G）| + | E（G）| \），我们说G当平等时，贪婪是贪婪的。当一个图不能是贪婪的时，包含它作为诱导子图的任何图也不能是贪婪的。图是关键的当它不是sp-greedy，但是其所有适当的诱导子图都是sp-greedy时。我们证明了无限的sp临界图族的存在。作为推论，我们证明sp贪婪图或sc贪图图都不能通过禁止有限子族来表征。