Linear discrepancy for vector-labelled perfect matchings
Year: 2026
Type: preprint
Abstract: Unrefereed research candidate. In one sentence: if the edges of a complete bipartite graph carry vector labels (for example colours), some perfect matching gets within 17k of any fractional target in any k-dimensional norm, improving the previous O(k2) bound to linear. Colour the edges of Kn,n with k colours and ask for a perfect matching that uses every colour in its expected proportion. Hogan, Scott and Tsarev (2026) showed that the total imbalance can be kept to O(k2), independently of n; earlier, Hollom gave a uniform bound of order 4k2. This paper proves a linear bound in a more general setting: if every edge carries a label in the unit ball of a k-dimensional real normed space and x is any fractional perfect matching, some perfect matching has label sum within 17k of the x-weighted sum. For arbitrary targets the order k is best possible in the worst case over norms (an ℓ1 example forces k/2). For the uniform target of the colour-balancing problem the best known lower bound is √(k/2), so that case remains open between √(k/2) and 17k. The key tool is a splitting lemma in the Hobby-Rice / necklace-splitting tradition: d signed measures on an interval cut into q pieces, with a different fraction wanted on each piece, can be matched by one set with at most q-1+2d switches. Laying the leftover fractional paths end to end on a line and cutting them all at once gives the rounding. Consequences: linear-in-k versions of colour-balanced perfect matchings of complete graphs, spanning forests, Hamilton cycles, bounded-degree spanning graphs and complete multipartite hypergraph matchings, and an existence version of a school-assignment rounding theorem of Santhini, Munagala, Nasre and Sankar (FORC 2025) with total capacity violation at most 8g-1 for g groups instead of O(g2). The proofs are existential; no efficient algorithm is claimed and the constant 17 is not optimised. The archive contains the manuscript (PDF and Markdown source), Figure 1 and the script that draws it, an exact checker (check_lemmas.py, Python standard library only, 2,374,625 integer and rational checks of the finite bookkeeping, with a mutation self-test) and its retained output, the bibliography with its Crossref, DataCite and arXiv records, the novelty report, a revision note, CI and SHA-256 sums. Assurance: written proofs; the checker verifies finite bookkeeping, not the Hobby-Rice step, the compactness argument or any statement for all n and k; internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text and figures CC0-1.0; code MIT.
Author: Anonymous
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225195994