Dinis Vitorino | Searcharxiv

arXiv2026

Bipartite Graphs Are Not Well-Quasi-Ordered by Bipartite Minors

In "Bipartite minors" [Journal of Combinatorial Theory, Series B, 2016], Chudnovsky et al. introduced the bipartite minor relation, a quasi-order on the class of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-quasi-order. We answer this question negatively by giving an infinite set of 2-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs G, H such that H is a bipartite minor, but not a minor, of G, and one set of pairs G, H such that H is a minor, but not a bipartite minor, of G.

math.CO↗

arXiv2025

Some Bounds Related to the $2$-adic Littlewood Conjecture

For every irrational real $α$, let $M(α) = \sup_{n\geq 1} a_n(α)$ denote the largest partial quotient in its continued fraction expansion (or $\infty$, if unbounded). The $2$-adic Littlewood conjecture (2LC) can be stated as follows: There exists no irrational $α$ such that $M(2^k α)$ is uniformly bounded by a constant $C$ for all $k\geq 0$. In 2016, Badziahin proved (considering a different formulation of 2LC) that if a counterexample exists, then the bound $C$ is at least $8$. We improve this bound to $15$. Then we focus on a ``B-variant'' of 2LC, where we replace $M(α)$ by $B(α) = \limsup_{n\to \infty} a_n(α)$. In this setting, we prove that if $B(2^k α) \leq C$ for all $k\geq 0$, then $C \geq 5$. For the proof we use Hurwitz's algorithm for multiplication of continued fractions by 2. Along the way, we find families of quadratic irrationals $α$ with the property that for arbitrarily large $K$ there exist $β, 2β, 4 β, \ldots, 2^K β$ all equivalent to $α$.

math.NT↗