On endomorphism universality of sparse graph classes

Solving a problem of Babai and Pultr from 1980 we show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a graph of bounded degree. Indeed we show that maximum degree $3$ suffices, which is best-possible. On the way we generalize a classic result of Frucht by showing that every group is the endomorphism monoid of a graph of maximum degree $3$ and we answer a question of Ne\v{s}et\v{r}il and Ossona de Mendez from 2012, presenting a class of bounded expansion such that every monoid is the endomorphism monoid of one of its members. On the other hand we strengthen a result of Babai and Pultr and show that no class excluding a topological minor can have all completely regular monoids among its endomorphism monoids. Moreover, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids.Comment: 37 pages, 18 figure