On graphs with a given endomorphism monoid

Hedrlín and Pultr proved that for any monoid M there exists a graph G with endomorphism monoid isomorphic to M. In this paper we give a construction G(M) for a graph with prescribed endomorphism monoid M. Using this construction we derive bounds on the minimum number of vertices and edges required to produce a graph with a given endomorphism monoid for various classes of finite monoids. For example we show that for every monoid M, |M|=m there is a graph G with End(G)≃M and |E(G)|≤(1 + 0(1))m2. This is, up to a factor of 1/2, best possible since there are monoids requiring a graph with \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}\begin{eqnarray*} && \frac{m^{2}}{2}(1 -0(1)) \end{eqnarray*}\end{document} edges.