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Statement

Setting (p. 8). For a loom $\mathbb L=(A,B)$ the paper writes $\tau^*(\mathbb L)=\tau^*(A\cup B)$. By its Corollary 2.5, every $(r,s)$-loom has $\tau^*(\mathbb L)\leq\max(r,s)$. Looms are defined in Definition 1.5.

Conjecture 4.1 (p. 8). If $\mathbb L=(A,B)$ is an $(r,s)$-loom, then $\tau^*(\mathbb L)=\max(r,s)$.

Conjecture 4.2 (p. 8). If $\mathbb L=(A,B)$ is an $(r,s)$-loom, then $\tau^*(A)=s$ and $\tau^*(B)=r$.

The paper notes (p. 8) that for $r=s$, Conjecture 4.1 would give $\max(\tau^*(A),\tau^*(B))=r$ by Theorem 2.6, and presents Conjecture 4.2 as the more general statement. By Lemma 1.12, Conjecture 4.2 says that both components of the loom have perfect fractional matchings, the question the paper raises on p. 4.

Proposition 4.3 (p. 8). If Conjecture 4.2 is true for an $(r,s)$-loom $(A,B)$, then $|V(A)|=|V(B)|=rs$.

The paper deduces (p. 8) that Conjecture 4.2 implies the Gyárfás--Lehel conjecture (its Conjecture 1.2, quoted on the Definition 1.5 page): a counterexample may be taken to be an $r$-partite $(r,r)$-loom, which would then have $r^2$ vertices, so some side of the partition has at most $r$ vertices and is a cover of $A\cup B$.

Proved cases recorded in the paper: $s=2$ by Theorem 7.3; $r=s=3$ by Corollary 8.2; looms of the form $(PM(G),C_s(PM(G)))$ for an $s$-regular graph $G$ by Theorem 6.4; and blow-ups of looms that satisfy it, under the hypotheses of Theorem 5.10, by Corollary 5.11 (p. 14).

Proof pointer

Proposition 4.3, p. 8: a fractional matching of $A$ of weight $s$ is perfect by Lemma 1.12, and double counting its weight over the vertices gives $|V(A)|=rs$.

Read depth

Claims checked: Conjectures 4.1 and 4.2, Proposition 4.3 and the deduction of Conjecture 1.2 were read clause by clause on the print. Nothing here is independently reviewed.

Dependencies

Definition 1.5 and Lemma 1.12 of the paper.

Source. R. Aharoni, E. Berger, J. Briggs, H. Guo and S. Zerbib, Looms, Discrete Math. 347 (2024), no. 12, 114181, arXiv:2309.03735; the edition read is named on the source card.

Bears on

None of the problem pages directly.