Explicit 3-colorings for exponential graphs

08/27/2018

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by   Adrien Argento, et al.

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Let H=(V,E) denote a simple, undirected graph. The 3-coloring exponential graph on H is the graph whose vertex set corresponds to all (not necessarily proper) 3-colorings of H. We denote this graph by K_3^H. Two vertices of K_3^H, corresponding to colorings f and g of H, are connected by an edge in K_3^H if f(i) ≠ g(j) for all ij ∈ E. El-Zahar and Sauer showed that when H is 4-chromatic, K_3^H is 3-chromatic el1985chromatic. Based on this work, Tardif gave an algorithm to (properly) 3-color K_3^H whose complexity is polynomial in the size of K_3^H tardifAlg. Tardif then asked if there is an algorithm in which the complexity of assigning a color to a vertex of K_3^H is polynomial in the size of H. We present such an algorithm, answering Tardif's question affirmatively.