On endomorphism universality of sparse graph classes

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 Nešetřil 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).

Figures (18)

• Figure 1: Classes of monoids.
• Figure 2: Sparse classes of graphs. Green stands for endomorphism universality and violet for non-universality results.
• Figure 3: The Hedrlín--Pultr graph $H^k$.
• Figure 4: A single-colored $c$-arc $a=(x,y)$ in $D$ and its replacement in $\check{D}_k$. The shortest cycles of $H^k_{c,a}$ are labelled $Z'_1,Z'_2,Z'_3$ in correspondence with those of $H^k$.
• Figure 5: Left: a vertex $v$ of $D$ with its outgoing and ingoing arcs and the labels of the colors. Right: the vertices of the walk $W_v$, their $0$-neighbours, and their ingoing and outgoing $K$-arcs in $D'$, assuming that $K=\{1,2,3\}$. The arcs in $A'_0$ are orange, the arcs in $A'_{c^+}$ are dashed, the arcs in $A'_{c^-}$ are dotted, and the arcs in $A'_c$ are solid.

Theorems & Definitions (94)

• Lemma 2.1
• Lemma 3.1
• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Proposition 3.6
• proof