arXiv subjects

Jitendra Prajapati

Publications and source records attributed to Jitendra Prajapati.

4 recordsLinked to original sources

arXiv2026

Coven-Meyerowitz T2 necessity through coprime stripe collapse

We prove that every finite subset of the integers which tiles by translations satisfies the Coven-Meyerowitz condition T2, with no restriction on the number of prime factors or their exponents. Together with the necessity of T1 and the sufficiency of T1 and T2 proved by Coven and Meyerowitz, this gives their proposed characterization of finite integer tiles. The proof uses strong induction on a cyclic tiling period. Character identities produce periodic Boolean product stripes; integral descent to a coprime quotient and the Frobenius identity force a common orientation. Independent phase shifts then give smaller-period tilings from which the mixed cyclotomic zeros of the original factors can be recovered. A companion Lean formalization verifies the unrestricted T2 necessity statement.

math.GM↗

arXiv2026

A counterexample to the claw-free Schur-positivity conjecture

The claw-free Schur-positivity conjecture, recorded by Stanley (1998) and credited there to Gasharov, asserts that the chromatic symmetric function of every claw-free graph is Schur-positive. We give a counterexample on 12 vertices: the line graph $G$ of the graph obtained from a 4-cycle by attaching triangles at two opposite vertices and pendant edges at the other two satisfies $[s_{(3,3,3,3)}]X_G = -64$. The coefficient follows from a short computation by hand and is also reproduced by three exact implementations. An exhaustive computation over all 216,777 connected claw-free graphs on at most 11 vertices shows that every one is Schur-positive, so 12 vertices is the minimum order of any counterexample. A complete census of the 1,728,404 connected claw-free graphs on 12 vertices finds exactly two non-Schur-positive isomorphism classes; the other has graph6 code K?`CR@`bAbRB and coefficient $[s_{(3,3,3,3)}] = -40$.

math.CO↗

arXiv2026

An explicit construction of two completely independent spanning trees in the four-dimensional dual-cube

Lalou, Mbarek, Skender and Togni (arXiv:2607.25917) proved that the $n$-dimensional dual-cube $F_n$ admits two completely independent spanning trees for every $n\ge 5$, observed that none exist for $n\le 3$, and identified $F_4$ as the first unresolved case, reporting more than 700 hours of inconclusive computation. We settle this case affirmatively by an explicit construction, completing the classification: $F_n$ admits two completely independent spanning trees if and only if $n\ge 4$. The internal-vertex sets of the two trees are the level sets of a single ten-term cubic polynomial over $\mathbb{F}_2$ in the seven vertex bits, and correctness reduces to finite connectivity checks that are machine-verified by a solver-free program distributed with the certificate. In $F_4$ the two trees necessarily use 254 of the 256 edges. We also report exact infeasibility results for simpler rules of the same shape: within the search model, no affine or quadratic rule works, and ten terms is the fewest possible for a cubic rule.

math.CO↗

arXiv2026

A counterexample to the Etzion-Silberstein conjecture

The Etzion-Silberstein conjecture asserts that the Singleton-type upper bound for linear Ferrers-diagram rank-metric codes is attained for every Ferrers diagram, minimum rank distance, and finite field. Let $E$ be the Ferrers diagram with column heights $(5,5,5,5,1,1)$. The bound for minimum rank distance $3$ is $12$. We prove that every binary linear code supported on $E$ with minimum rank distance $3$ has dimension at most $11$, and we give an explicit code of dimension $11$. Thus the optimum is exactly $11$, disproving the conjecture. The nonexistence proof reduces a hypothetical dimension-$12$ code to one of the three equivalence classes of binary $[4\times 4,12,2]$ MRD codes. A rank-distribution argument eliminates two classes and leaves four kernel orbits in the field class; all four exact lift systems are unsatisfiable. Independently written verifiers reproduce the result, including a raw enumeration of all $8,382,465$ kernels without orbit reduction. We also prove an exact row-cone propagation identity. Iterating it produces binary counterexamples with bound $12$ and optimum $11$ at every minimum rank distance $d \geq 3$.

cs.IT↗