Related papers: Triangle-free triple systems
In this study we consider the problem of triangulated graphs. Precisely we give a necessary and sufficient condition for a graph to be triangulated. This give an alternative characterization of triangulated graphs. Our method is based on…
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…
We study here the graphs with seven vertices in an effort to classify which of them appear as the prime character degree graphs of finite solvable groups. This classification is complete for the disconnected graphs. Of the 853…
New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in $\mathbb{P}^2\times \mathbb{P}^2$ are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with…
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic…
The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…
Considering regular graphs with every edge in a triangle we prove lower bounds for the number of triangles in such graphs. For r-regular graphs with r <= 5 we exhibit families of graphs with exactly that number of triangles and then…
We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is…
We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Bj\"orner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We…
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
The Burling sequence is a sequence of triangle-free graphs of increasing chromatic number. Each of them is isomorphic to the intersection graph of a set of axis-parallel boxes in $R^3$. These graphs were also proved to have other…
We present a new construction of triple arrays by combining a symmetric 2-design with a resolution of another 2-design. This is the first general method capable of producing non-extremal triple arrays. We call the triple arrays which can be…
We classify spherical quadrilaterals up to isometry in the case when one inner angle is a multiple of pi while the other three are not. This is equivalent to classification of Heun's equations with real parameters and one apparent…
We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…
We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite…
Steinberg and Tovey proved that every n-vertex planar triangle-free graph has an independent set of size at least (n+1)/3, and described an infinite class of tight examples. We show that all n-vertex planar triangle-free graphs except for…
Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a…
An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is the maximum number of edges in an…
Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid…
Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao has shown recently that the maximum possible…