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I saw a recent question from M.R. and realized there is no function to compute the chromatic index and number of a graph, other than a really slow method in the now deprecated Combinatorica.

So, how can I compute the chromatic index and number of a graph?

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You can compute the chromatic index of a graph by first observing it is equivalent to the chromatic number of the line graph of the graph. This immediately suggests straightforward algorithms:

ChromaticNumber[g_] := MinValue[{z, z > 0 && ChromaticPolynomial[g, z] > 0}, z, Integers];
ChromaticIndex[g_] := ChromaticNumber[LineGraph[g]];

Notice we make use of ChromaticPolynomial which was incorporated into the main language in V10.

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