On rigid regular graphs and a problem of Babai and Pultr

TL;DR

The paper addresses the problem of rigidity in finite graphs by proving that for every $d\ge 3$, there exist infinitely many mutually rigid $d$-regular graphs of odd girth $g\ge 7$ and by determining the minimal order $\nu(d)$ of a rigid $d$-regular graph. It introduces a comprehensive toolkit combining directed gadgetry, the šíp product, and tiling factors to construct large families of rigid graphs with prescribed odd girth, and it shows how to build $d$-indicators $S(d,g)$ to realize these properties. A central achievement is representing every finite monoid as the endomorphism monoid of a regular graph, solving a Babai–Pultr problem, via a two-stage process that first handles binary relational systems and then transfers to graphs while preserving endomorphism structure. The results have broad implications for endomorphism universality, graph symmetry, and the design of graphs with tailored endomorphism monoids, offering constructive methods to realize monoids within regular graphs and guiding future inquiries into rigidity and monoid representations in graph theory.

Abstract

A graph is \textit{rigid} if it only admits the identity endomorphism. We show that for every $d\ge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $g\geq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $d\ge 3$. This provides strong positive answers to a question of van der Zypen [https://mathoverflow.net/q/296483, https://mathoverflow.net/q/321108]. Further, we use our construction to show that every finite monoid is isomorphic to the endomorphism monoid of a regular graph. This solves a problem of Babai and Pultr [J. Comb.~Theory, Ser.~B, 1980].

Figures (10)

• Figure 1: A smallest $3$-regular rigid graph with g6-code MCHYe??KOCBOC?g_ and the subdivided graph $G_r$ from Lemma \ref{['lem:cubic']}.
• Figure 2: $S(4,3,f)$ for $f$ given by $(i,j)$$(1,3)$$(1,2)$$(2,3)$$(2,2)$$(3,3)$$(3,2)$$f(i,j)$$-$$-$$+$$0$$0$$-$.
• Figure 3: An indicator $S$ with distinguished pair of vertices $(\mathrm{in}\,S,\mathrm{out}\,S)=(i,o)$, an arc $a=(x,y)$ in a binary $\{1\}$-system (i.e. a digraph) $D$, and the gadgets replacing $a$ in $D\mathbin * S$ and in $D\mathbin{\vec{*}} S$.
• Figure 4: A $7$-tiling graph.
• Figure 5: The two graphs $T$ (left) and $T'$ (centre) of the $g$-tiling factor $\mathbf T=(T,T')$ from a $g$-tiling graph $G$, and the graph $\overline T$ from Lemma \ref{['lem:tiling_factor_girth']} (right), obtained by adding to $T$ the edge $\{u_0,u_1\}$ and all the edges of $T'$. In the illustration we assume that the symmetries of the ellipse correspond to the symmetries of $G$. Only the vertices from $U$ (larger) and some other vertices of the border (smaller) have been depicted. Twisted edges appear in red, and straight edges in dark blue.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Claim 3.4.1