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Erdos theorem says it is possible and it is not so easy. What is the general procedure to construct graphs like Grötzsch graph?

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    $\begingroup$ I do not see any research aspect to this question. Hence it belongs at math.stackexchande, not here. $\endgroup$
    – Chris Godsil
    Commented Dec 19, 2014 at 3:56

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Two explicit constructions as certain Cayley graphs of $\operatorname{PGL}(\mathbb F_q)$ and $\operatorname{PSL}(\mathbb F_q)$ are detailed in chapters 3 and 4 of the book "Elementary Number Theory, Group Theory, And Ramanujan Graphs" by Giuliana Davidoff, Peter Sarnak and Alain Valette.

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