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Precoloring extension for $$K_4$$-minor-free graphs. (English) Zbl 1229.05155
Summary: Let $$G=(V, E)$$ be a graph where every vertex $$v\in V$$ is assigned a list of available colors $$L(v)$$. We say that $$G$$ is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If $$L(v)=\{1,\dots , k\}$$ for all $$v\in V$$ then a corresponding list coloring is nothing other than an ordinary $$k$$-coloring of $$G$$. Assume that $$W\subseteq V$$ is a subset of $$V$$ such that $$G[W]$$ is bipartite and each component of $$G[W]$$ is precolored with two colors taken from a set of four. The minimum distance between the components of $$G[W]$$ is denoted by $$d(W)$$. We will show that if $$G$$ is $$K_4$$-minor-free and $$d(W)\geq 7$$, then such a precoloring of $$W$$ can be extended to a 4-coloring of all of $$V$$. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of $$G$$ for outerplanar graphs, provided that $$|L(v)|=4$$ for all $$v\in V\setminus W$$ and $$d(W)\geq 7$$. In both cases the bound for $$d(W)$$ is best possible.

##### MSC:
 05C35 Extremal problems in graph theory 05C05 Trees 05C83 Graph minors 05C38 Paths and cycles 05C15 Coloring of graphs and hypergraphs
##### Keywords:
precoloring extension; list coloring; minor-free graphs
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