A counter-example to Baranyai's combinatorial characterisation for 3-rigidity

TL;DR

The paper addresses the challenge of a combinatorial characterisation for $3$-dimensional graph rigidity by presenting a counter-example to the converse of Baranyai's necessary condition and by supplying an entirely linear-algebraic proof of the condition. It shows that the double banana graph is $(3,6)$-tight yet not $3$-rigid, thereby refuting the claimed converse, and provides a self-contained algebraic proof that, for minimally $3$-rigid graphs, the stated partitions ensuring $2$-rigidity exist. Additionally, it discusses how coning might extend the counter-example to higher dimensions and proposes a refined conjecture for $(3,6)$-tight graphs based on edge-wise decompositions into $(2,3)$-tight subgraphs. Overall, the work clarifies limitations of existing combinatorial criteria for 3D rigidity and suggests directions for future, dimensionally extending results in rigidity theory.

Abstract

Recently Baranyai described a necessary combinatorial characterisation of graph rigidity for dimension 3. In this short note we provide a counter-example to the converse of the condition. Additionally, we provide an alternative proof to the Baranyai's necessary condition.

Theorems & Definitions (11)

• Theorem 1.1: Baranyai baranyai2026genericrigiditygraphs
• Theorem 2.1: Geiringer-Laman theorem pollaczekgeiringer1927Laman1970
• Lemma 4.1
• Lemma 4.2
• proof
• Claim 1
• proof
• Lemma 4.3
• proof
• proof : Proof of \ref{['mainthm']}