On rigid regular graphs and a problem of Babai and Pultr

2025-02-17

AI Summary

This paper addresses two fundamental problems: (1) constructing infinitely many pairwise non-isomorphic rigid $d$-regular graphs for any degree $d geq 3$ and odd girth $g geq 7$, and precisely determining their minimum order; (2) proving that every finite monoid is isomorphic to the endomorphism monoid of some regular graph—fully resolving the Babai–Pultr (1980) problem and the van der Zypen conjecture. Methodologically, the work integrates explicit graph constructions, structural analysis of graph endomorphisms, and techniques from combinatorial and algebraic graph theory, establishing—for the first time—a systematic characterization of existence and minimality for rigid regular graphs. Key contributions include: a closed-form formula for the minimum order; a surjective realization of arbitrary finite monoids as endomorphism monoids of regular graphs; and a novel correspondence paradigm linking graph rigidity with algebraic structure.

Technology Category

Knowledge Representation and Reasoning: Nonmonotonic ReasoningReasoning under Uncertainty: Graphical ModelsData Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphsSemantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologiesWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A graph is extit{rigid} if it only admits the identity endomorphism. We show that for every $dge 3$ there exist infinitely many mutually rigid $d$-regular graphs of arbitrary odd girth $ggeq 7$. Moreover, we determine the minimum order of a rigid $d$-regular graph for every $dge 3$. 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].

Problem

• Existence of infinitely many rigid d-regular graphs.
• Determination of minimum order for rigid d-regular graphs.
• Endomorphism monoid isomorphism for every finite monoid.

Innovation

• Constructs infinitely rigid graphs
• Determines minimal graph orders
• Links monoids to graph endomorphisms