On endomorphism universality of sparse graph classes

TL;DR

This paper proves that every commutative idempotent monoid can be realized as the endomorphism monoid of a subcubic graph, and explores limitations of such representations within minor-closed classes, with implications for graph classes and monoid representations.

Contribution

It establishes the universality of subcubic graphs for commutative idempotent monoids and explores constraints on representing monoids within minor-closed graph classes.

Findings

Every commutative idempotent monoid is the endomorphism monoid of a subcubic graph.

No class excluding a minor can have all commutative idempotent monoids as endomorphism monoids.

Monoids can be represented by graphs of bounded expansion and bounded degree.

Abstract

We show that every commutative idempotent monoid (a.k.a lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980] and the degree bound is best-possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by-product we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Ne\v{s}et\v{r}il and Ossona de Mendez) and $k$-cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).