相关论文: On a result concerning algebraic curves passing th…

A recent result of Bokal et al. [Combinatorica, 2022] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is c=13. The key to that result is an inductive construction of a family…

The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known…

A planar node set $\mathcal X,$ with $\#\mathcal X=\binom{n+2}{2},$ is called $GC_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line if the line is a factor of the…

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of…

We prove that for any linear 3-graph on $n$ vertices without a path of length 5, the number of edges is at most $\frac{15}{11}n$, and the equality holds if and only if the graph is the disjoint union of $G_0$, a graph with 11 vertices and…

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed field $k$. A base point free complete very special linear system $g^r_n$ on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor $E$…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

Ihara's proof that the reduction of the modular curve $X_0(n)$ at a prime $p$ not dividing $n$ has many points over a quadratic extension is adapted to the drinfeld modular curves $X_0(n)$. In order to do so, some properties of drinfeld…

We describe symmetries of the braid monodromy decomposition for a class of plane curves defined over reals including the real curves with no real points and proving new divisibility relations for Alexander invariants of such curves.

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field.…

Since the works of Krasnov and Scheiderer, there has been an interest in studying effective totally real divisors on a curve X defined over a real closed field, i.e., effective divisors supported on the real locus. Scheiderer proved that,…

Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this setup, characterization and classification problems lead to the natural question: "Given two points on X, how…

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known…

Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then…

In the present note we construct new families of free plane curves starting from a curve $C$ and adding high order inflectional tangent lines of $C$, lines joining the singularities of the curve $C$, or lines in the tangent cone of some…

In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…

Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is…

We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes. We prove that if $\mathcal X$ is a superelliptic curve defined over…

We consider numbers and sizes of independent sets in graphs with minimum degree at least $d$, when the number $n$ of vertices is large. In particular we investigate which of these graphs yield the maximum numbers of independent sets of…