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# Bounds on the Dynamic Chromatic Number of a Graph in Terms of its Chromatic Number

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A vertex coloring of a graph is called dynamic if the neighborhood of any vertex of degree at least 2 contains at least two vertices of distinct colors. Similarly to the chromatic number χ(G) of a graph G, one can define its dynamic number χd(G) (the minimum number of colors in a dynamic coloring) and dynamic chromatic number χ2(G) (the minimum number of colors in a proper dynamic coloring). We prove that χ2(G) ≤ χ(G) · χd(G) and construct an infinite series of graphs for which this bound on χ2(G) is tight.

For a graph G, set $$k=\left\lceil \frac{2\Delta (G)}{\delta (G)}\right\rceil$$ We prove that χ2(G) ≤ (k+1)c. Moreover, in the case where k ≥ 3 and Δ(G) ≥ 3, we prove the stronger bound χ2(G) ≤ kc.

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