Graphs and Hypergraphs
| Hi-index | 0.04 |
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 χ ≤ 3, and (χ, p) = (4, 1) or (4, 2).