Excess Obstructions and Star-Isolated Certificates for the Hypergraph Nash--Williams--Tutte Conjecture

TL;DR

This paper disproves a conjecture in hypergraph theory related to the Nash--Williams--Tutte conjecture, identifies the precise structural condition needed, and constructs specific counterexamples and positive instances.

Contribution

It reveals the conjecture's literal form is false, clarifies the necessary condition for the hypergraph property, and provides a detailed structural analysis and construction of critical cases.

Findings

The conjecture is false in its original form due to excess edge count.

A corrected form of the conjecture is identified with an exact equality condition.

Constructs a large class of non-graphic critical positive instances using layer-contained star realizations.

Abstract

Guo, Li, Shangguan, Tamo, and Wootters formulated in SIAM Journal on Computing a hypergraph Nash--Williams--Tutte conjecture: every $k$-weakly-partition-connected hypergraph on $t$ vertices should admit a $k$-distinguishable tree assignment. We show that the conjecture, in its literal published form, is false for a sharp and structural reason. A tree assignment replaces every hyperedge $e$ by a tree with $|e|-1$ labelled edges, so its edge number is the excess $\rho(H)=\sum_e(|e|-1)$. A $k$-tree decomposition, however, has exactly $k(t-1)$ edges. Thus $\rho(H)=k(t-1)$ is a necessary condition, whereas weak partition connectivity only implies $\rho(H)\ge k(t-1)$. Consequently, for every $t\ge2$, $k\ge1$, and $q\ge1$, the hypergraph consisting of $k+q$ copies of the full hyperedge $V$ is $k$-weakly-partition-connected but has no $k$-distinguishable tree assignment. We then isolate the critical corrected form, prove that its equality is exactly the equality required for the full intersection-matrix row set, and give a large non-graphic class of critical positive instances. The positive construction uses layer-contained star realizations and extremal signature weights, producing weak partition connectivity by a quotient-rank argument and unique signatures under one-vertex sums and explicit two-sided star blocks.