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Ordered colourings. (English) Zbl 0842.05032
An ordered \(k\)-coloring of a graph \(G\) is a coloring function \(c: V(G)\to \{1, 2,\dots, k\}\) such that, for every pair of distinct vertices \(x\) and \(y\) and for every \(x\)-\(y\) path \(P\), if \(c(x)= c(y)\), then there exists an internal vertex \(z\) of \(P\) such that \(c(x)< c(z)\). This paper proves some results about ordered colorings of trees and planar graphs. For example, if every planar graph has an ordered coloring using at least \(g(v)\) vertices, then \(g(v)\leq 3(\sqrt 6+ 2)\sqrt v\). The paper also examines the relationship between connectivity and ordered colorings.

MSC:
05C15 Coloring of graphs and hypergraphs
05C05 Trees
05C10 Planar graphs; geometric and topological aspects of graph theory
05C35 Extremal problems in graph theory
05C40 Connectivity
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