arXiv subjects

Sanjit Singh Mehat

Publications and source records attributed to Sanjit Singh Mehat.

3 recordsLinked to original sources

arXiv2026

Albertson's Conjecture for Chromatic Numbers at Most 29

Albertson's conjecture asserts that every finite simple graph $G$ with $χ(G) \ge r$ satisfies $\operatorname{cr}(G) \ge \operatorname{cr}(K_r)$. Building on Cranston's verification for $r \le 24$ and his reduction of $r \in \{25,26\}$ to three residual orders, we eliminate those residual cases and then prove the cases $r=27,28,29$. The first structural ingredient is a Kempe-chain construction: if a $k$-critical graph has a vertex of degree $k-1$, then it contains a branch-clean essential immersion of $K_k$. Essential immersions are crossing-monotone, so a critical counterexample must have minimum degree at least $k$. For $r=27$, this one-unit degree gain, Gallai's join structure, critical-graph edge bounds, and induced-subgraph averaging close every possible order. For $r=28$ and $r=29$, the remaining near-$2r$ orders are converted to dense complements. Stehlík's coloring theorem makes the odd-order complements factor-critical; a clique-partition obstruction yields an anti-tight matching property; and Tutte barriers, Hall-type expansion, and deficit bookkeeping eliminate the final cases. At order 58 for $r=29$, Rabern's coloring inequality handles the regular case, while the last degree-deficit-two case is reduced to two disjoint triangles and a finite barrier analysis.

math.CO↗

arXiv2026

Tensor constructions for Euler magic matrices and proper examples of orders 9, 27, 81 and 243

An Euler magic matrix is an integer matrix $M$ satisfying $MM^{\mathsf{T}}=γI$ together with two diagonal square-sum conditions; it is proper when its entry squares are pairwise distinct. M{ü}ller proved that Euler magic matrices exist in every order other than $3$, and that no Euler magic matrix of order $3$ exists at all, while proper examples are considerably more restrictive. We describe a tensor construction whose factors are only required to satisfy the orthogonality equation $AA^{\mathsf{T}}=γI$: for a linear reindexing $L$ of the row index group $\mathbb{F}_3^k$ obeying an explicit support condition, the reindexed Kronecker product of $k$ such $3\times3$ factors satisfies the full Euler magic conditions in order $3^k$. Choosing factors whose entry squares have pairwise distinct products, we obtain proper Euler magic matrices of orders $9$, $27$, $81$ and $243$. The construction therefore produces proper examples in powers of three even though order three admits no Euler magic matrix, and the passage from the factors to the product is exactly where the Euler conditions are created rather than inherited. We give an explicit support condition on $L$ and show that it characterises the linear reindexings forcing the two Euler diagonal identities for every tuple of semi-magic factor arrays; over $\mathbb{F}_p$ with two factors, such a reindexing exists only when $p\le3$. The four existence results are formalised in Lean 4, as are the two instances of the construction used to obtain them; the order-$243$ certificate is taken from the archived development and was not rebuilt in preparing this paper, although its witness was reproduced here by exact integer arithmetic. Further proper examples of orders $729$ and $2187$ are verified by exact integer computation only.

math.GM↗

arXiv2026

A proper Euler magic matrix of order 6

An Euler magic matrix is an integer matrix M with MM^t = gamma I for some gamma != 0, whose squared entries sum to gamma along both main diagonals; it is proper if its squared entries are pairwise distinct. Euler gave a proper example of order 4; Müller settled orders 3 (none exists) and 8; and Kominers settled order 5. We give an order-6 construction, a case not addressed by Müller or Kominers, exhibiting a proper Euler magic matrix of order 6 with gamma = 18500 together with a second, independent one with gamma = 43290. The proof is the explicit matrix and a finite exact verification. We also record an elementary counting bound gamma >= 2485 for proper order-6 examples.

math.GM↗