Note
Locally restricted colorings

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Abstract

We consider the following problem: given suitable integers χ and p, what is the smallest value ρ such that, for any graph G with chromatic number χ and any vertex coloring of G with at most χ+p colors, there is a vertex v such that at least χ different colors occur within distance ρ of v? Let ρ(χ,p) be this value; we show in particular that ρ(χ,p)p/2+1 for all χ,p. We give the exact value of ρ when p=0 or χ3, and (χ,p)=(4,1) or (4,2).

Keywords

Chromatic number
Graph coloring
Locally restricted colorings

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Supported by Grant no. 2100-068172.02 of the Swiss National Science Foundation.