## 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

### Keywords:

ordered colorings; trees; planar graphs; connectivity
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### References:

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