相关论文: Slicing all Edges of an $n$-cube Requires $n^{2/3}…

We consider the space $[0,n]^3$, imagined as a three dimensional, axis-aligned grid world partitioned into $n^3$ $1\times 1 \times 1$ unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it,…

We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal…

Motivated by its relation to the length of cutting plane proofs for the Maximum Biclique problem, we consider the following communication game on a given graph G, known to both players. Let K be the maximal number of vertices in a complete…

We show that any n-vertex complete graph with edges colored with three colors contains a set of at most four vertices such that the number of the neighbors of these vertices in one of the colors is at least 2n/3. The previous best value,…

In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded degree…

Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

Given a hypergraph $\mathcal{H}$, we introduce a new class of evaluation toric codes called edge codes derived from $\mathcal{H}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the…

Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges have…

Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) :=…

It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$…

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…

While the problem of determining whether an embedding of a graph $G$ in $\mathbb{R}^2$ is {\it infinitesimally rigid} is well understood, specifying whether a given embedding of $G$ is {\it rigid} or not is still a hard task that usually…

In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least…

We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded…

The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…

A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…

The hypercube of dimension n is the graph whose vertices are the 2^n binary words of length n, and there is an edge between two of them if they have Hamming distance 1. We consider an edit distance based on swaps and mismatches, to which we…

A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote…

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition…