Related papers: New Upper Bound for the Edge Folkman Number Fe(3,5…

An adjacency-crossing graph is a graph that can be drawn such that every two edges that cross the same edge share a common endpoint. We show that the number of edges in an $n$-vertex adjacency-crossing graph is at most $5n-10$. If we…

A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is $k$ and $G$ has only one $k$-coloring up to permutation of the colors. A uniquely $k$-colorable graph $G$ is edge-critical if $G-e$ is not a uniquely $k$-colorable…

Let $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such…

For a given graph $H$, the Ramsey number $r(H)$ is the minimum $N$ such that any 2-edge-coloring of the complete graph $K_N$ yields a monochromatic copy of $H$. Given a positive integer $n$, a \emph{fan }$F_n$ is a graph formed by $n$…

In this note we obtain a new bound for the acyclic edge chromatic number $a'(G)$ of a graph $G$ with maximum degree $D$ proving that $a'(G)\leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis,…

Let $f(n, v, e)$ denote the maximum number of edges in a 3-uniform hypergraph on $n$ vertices which does not contain $v$ vertices spanning at least $e$ edges. A central problem in extremal combinatorics, famously posed by Brown, Erd\H{o}s…

Floating edge bands (FEB) have been identified in systems such as obstructed atomic insulators and layered nonsymmorphic semimetals, attracting considerable interest recently. Here we demonstrate that FEB can arise in a simplified model…

We consider spanning trees of $n$ points in convex position whose edges are pairwise non-crossing. Applying a flip to such a tree consists in adding an edge and removing another so that the result is still a non-crossing spanning tree.…

The no-(k+1)-in line problem seeks the maximum number of points that can be selected from an $n \times n$ square lattice such that no $k+1$ of them are collinear. The problem was first posed more than $100$ years ago for the special case…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

Let $G$ be a bipartite graph without loops and multiple edges on $v\ge 4$ vertices, which can be drawn on the plane such that any edge intersects at most one other edge. We prove that such graph has at most $3v-8$ edges for even $v\ne 6$…

We revisit here a fundamental result on planar triangulations, namely that the flip distance between two triangulations is upper-bounded by the number of proper intersections between their straight-segment edges. We provide a complete and…

In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach (2017) and Ellenberg and Gijswijt (2017)), the classical cap set constructions had not been affected. In this work, we introduce a very…

We give a short proof of the following theorem due to Jon H. Folkman (1969): The chromatic number of any graph is at most $2$ plus the maximum over all subgraphs of the difference between half the number of vertices and the independence…

Let STS(n) denote the number of Steiner triple systems on n vertices, and let F(n) denote the number of 1-factorizations of the complete graph on n vertices. We prove the following upper bound. STS(n) <= ((1 + o(1)) (n/e^2))^(n^2/6) F(n) <=…

For Artin-Schreier curve y^q -y = f(x) defined over a finite field F_q of q elements, we show that the Weil bound for the number of the rational points over extension fields of F_q can often be greatly improved, essentially removing an…

Since 2002, the best known upper bound on the Ramsey numbers R n (3) = R(3,. .. , 3) is R n (3) $\le$ n!(e -- 1/6) + 1 for all n $\ge$ 4. It is based on the current estimate R 4 (3) $\le$ 62. We show here how any closing-in on R 4 (3)…

We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(q) have 14 edges. We give six examples which fall into two classes. One…

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

Let $M^m$ be an $m$-dimensional, smooth and closed manifold, equipped with a smooth involution $T\colon M^m \to M^m$ fixing submanifolds $F^n$ and $F^4$ of dimensions $n$ and $4$, respectively, where $4<n<m$ and $F^n\cup F^4$ does not…