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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
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