arXiv subjects

Ankan Sadhu

Publications and source records attributed to Ankan Sadhu.

5 recordsLinked to original sources

arXiv2026

Every $k$-Quasi-Planar Graph Has Linearly Many Edges

For every fixed integer $k\ge2$, we prove that every simple graph on $n$ vertices drawn in the plane with no $k$ pairwise crossing edges has at most $C_kn$ edges, where $C_k$ depends only on $k$. We allow any finite number of proper crossings between two edges, including edges with a common endpoint. For $n\ge3$, removing a small disk around each vertex turns the drawing into a family of arcs on a sphere with $n$ boundary circles. Each arc joins two different boundary circles, and different edges have different unordered pairs of endpoint circles. The arcs are therefore essential and pairwise nonhomotopic, with endpoints allowed to move along their boundary circles during a homotopy. Any family of pairwise intersecting lifts to the universal cover projects to distinct graph edges that cross pairwise. Its size is consequently at most $k-1$. Beeker and Lazarovich's theorem on surface patterns then bounds the number of arcs. Their estimate is linear in the size of a triangulation, and an explicit triangulation of the $n$-holed sphere has linearly many edges and triangles.

math.CO↗

arXiv2026

A Proof of Albertson's Conjecture

Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of order close to r, we construct edge-disjoint routes while preserving a complete bipartite subgraph whose forced crossings compensate for those introduced when the immersion is smoothed into a drawing. For larger orders, we derive explicit edge bounds using independent-triple deletion and structural properties of critical graphs, then combine them with sampling inequalities that account for crossings within a maximum clique. The remaining numerical inequalities are certified using exact rational arithmetic, with a reproducible certificate and standalone Python verifier.

math.CO↗

arXiv2026

Beyond Liu's 0.382709 threshold for the union-closed sets conjecture

The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\mathrm{UC}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\mathrm L}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound $c_{\mathrm{UC}}\ge c_{\mathrm L}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\mathrm{UC}}\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\ldots$.

math.CO↗

arXiv2026

An improved upper bound for the Zarankiewicz number z(43;2) <= 294 via structural rigidity, including hexary packing number pa(5;6) = 31

The Zarankiewicz number z(n;2) is the largest number of ones in an n x n zero-one matrix with no all-one 2x2 submatrix. At n=43, the Kovari-Sos-Turan bound, sharpened by Reiman, gives z(43;2) <= 301 with no rounding slack; equality would force a projective plane of order six, which Tarry ruled out in 1900. That fact alone says nothing about how far below 301 the truth lies. We prove z(43;2) <= 294. The argument is local and geometric. Writing D = 301 - E for the deficiency of a configuration with E ones, a crossing count paired with its dual forces every line to at most eight points and every point to at most eight lines whenever D <= 6. A deficiency count then produces a clean point of degree seven, collinear with all others, unless the configuration has both a line of eight points and a point of degree eight. The seven lines through a clean point partition the remaining 42 points into groups that every other line meets at most once, so the number of (line, missed group) incidences equals D. At D=6 this forces group sizes 6,6,6,6,6,6,6 or 7,6,6,6,6,6,5; the second forces a TD(4,6), and the first yields 32 blocks of a partial TD(5,6), whose refutation needs an unknown packing number. We determine it. pa(5;6) and pa(6;6) are listed as open in the Handbook of Combinatorial Designs at 30 <= N <= 34; a class holds at most six words, so at most one full class gives at most 6 + 5*5 = 31, and what was missing was refuting the two-full-class case, done here by exhaustive search. Hence pa(5;6) = pa(6;6) = 31. With no clean point, the configuration is rigid enough to complete to a projective plane of order six. With a known 290-one configuration, this gives 290 <= z(43;2) <= 294. We also derive z(43;2) <= 299 via Totten's classification, and rule out configurations with 290+ ones from deleting rows/columns of PG(2,7).

math.CO↗

arXiv2026

A congruence obstruction to Roman's bound for Zarankiewicz numbers

Let z(m,n;s,t) be the largest number of ones in an m x n zero-one matrix with no s x t all-ones submatrix. Roman's 1975 inequality remains the best general upper bound for s>=3, but it is not attained on a large part of the range just below the design threshold T=(t-1)C(m,s)/(s+1). The proof has two steps. First, for n=T-c with 1<=c<=sT/(s+2), Roman's bound equals the elementary counting bound (s+1)(T-c)+floor(2c/s), adding nothing beyond a budget inequality and convexity. Second, attainment forces all but at most one column to have size s+1 or s+2; each such column has a point lying in a number of s-sets divisible by d=gcd(s,C(s+1,2)), pinning the leftover coverage there to a single residue mu mod d, which a global count rules out. With r=c mod s and slack sigma(r) depending only on s, we prove z(m,T-c;s,t) <= Rom(m,T-c)-1 whenever 1<=sigma(r) s*sigma(r). The first case is an odd-s phenomenon confined to one residue class, giving order-m^s values of n; the second needs mu != 0 but covers the whole interval once m exceeds a threshold depending only on s and mu. For s=4, t=2, m=28 it covers all 2730 values of n; when c=(s+1)/2 and an s-(m,s+1,t-1) design exists, z=(s+1)(T-c) exactly. Finally we relate the obstruction to linear programming: the relaxation over all 2^m subset variables collapses, under symmetrisation, to the counting bound, so no linear relaxation of the covering constraints alone can beat the bound of Chen, Horsley, and Mammoliti (arXiv:2310.12685, "Zarankiewicz numbers near the triple system threshold"). For the refined program of Davies, Gill, and Horsley, its optimum is still attained at the Roman vertex on an explicit sub-family, and we record where their program does better.

math.CO↗