bibkey: dalfofiolreyes2026threequarters authors: C. Dalfo and M. A. Fiol and M. A. Reyes year: 2026 title: A note on three-quarters circulant digraphs doi: 10.61091/um128-09 url: https://arxiv.org/html/2609.33718v1 claim: “Conjecture 2.4 proposes, for every k at least one, the lattice generated by (k+2,1-k) and (2,k), a three-quarters digraph of diameter k on k^2+4k-2 vertices with steps 1 and -k-4.” strata_touched:

• D5/S3/Arith/Covering/ThreeQuarterCirculantConjecture license: citation-only triage: anchor

A note on three-quarters circulant digraphs

Dalfó, Fiol, and Reyes define the three sector representations of a residue z using nonnegative m,n and exactly one of m*a+n*b, -m*a+n*b, and m*a-n*b modulo N. The sector distance is the least m+n in such a representation. This is narrower than allowing arbitrary mixtures of all positive and negative steps.

Conjecture 2.4 in section 2 states that, for every integer k >= 1, the lattice generated by (k+2,1-k) and (2,k) corresponds to a tile of a three-quarters digraph with diameter k. It proposes N=k^2+4k-2, a=1, and b=-k-4. The determinant of the displayed columns is N.

Section 1 calls CD(N,a,b) a regular degree-two digraph with generating set {a,b}. Section 2 defines a digraph’s arcs as a set of ordered pairs, and property P1 repeats that CD(N,a,b) is degree two. For the conjecture’s k=1 parameters, N=3 and -5=1 modulo three, so the generating set has one element and every vertex has one outgoing neighbor. The sector radius calculation at this parameter is a separate question from the degree-two graph assertion. Section 2 also describes a different k=1 example, TQ(5,1,2); that example does not change Conjecture 2.4’s specified parameters.

Verified locator

• Published source: Utilitas Mathematica 128 (2026), section 2, Conjecture 2.4; DOI 10.61091/um128-09.
• arXiv HTML version 1: https://arxiv.org/html/2609.33718v1 .
• Graph boundary: section 1 paragraph 3, section 2 paragraph 1 and P1.