Related papers: Cycles and Paths Embedded in Varietal Hypercubes
A $k$-uniform tight cycle is a $k$-graph with a cyclic order of its vertices such that every $k$ consecutive vertices from an edge. We show that for $k\geq 3$, every red-blue edge-coloured complete $k$-graph on $n$ vertices contains $k$…
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
Let $G$ be an edge-colored graph, a walk in $G$ is said to be a properly colored walk iff each pair of consecutive edges have different colors, including the first and the last edges in case that the walk be closed. Let $H$ be a graph…
Every convex polygon with $n$ vertices is a linear projection of a higher-dimensional polytope with at most $147\,n^{2/3}$ facets.
We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn…
A geometric graph is a drawing of a graph in the plane where the vertices are drawn as points in general position and the edges as straight-line segments connecting their endpoints. It is plane if it contains no crossing edges. We study…
We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone…
It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each…
Suppose that a finite-dimensional cube is orthogonally projected onto a central section of itself by a subspace of one dimension less. Up to dimension $9$, at least one vertex is projected onto the section, but for dimension $10$ or larger,…
A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard…
According to a computer search conducted by the author and described in [7], in $Q^+(6, 4)$ there are two types of hyperovals, having 72 and 96 points, respectively. Here we give geometric descriptions for these examples.
Cubical rectangles are being defined and explored here over the $n-$dimensional geometric cube $Q_n.$ They form a new class of geometric objects that includes all the edges and all the squares of the $n-$cube. We enumerate and characterize…
In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…
A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured…
Given a random k-dimensional cross-section of a hypercube, what is its expected number of vertices? We show that, for a suitable distribution of random slices, the answer is $2^k$, independent of the dimension of the hypercube.
We show that every $4$-chromatic graph on $n$ vertices, with no two vertex-disjoint odd cycles, has an odd cycle of length at most $\tfrac12\,(1+\sqrt{8n-7})$. Let $G$ be a non-bipartite quadrangulation of the projective plane on $n$…
We consider the problem of finding a Hamiltonian path or a Hamiltonian cycle with precedence constraints in the form of a partial order on the vertex set. We show that the path problem is $\mathsf{NP}$-complete for graphs of pathwidth 4…
Motivated by a relaxed notion of the celebrated Hamiltonian cycle, this paper investigates its variant, parity Hamiltonian cycle (PHC): A PHC of a graph is a closed walk which visits every vertex an odd number of times, where we remark that…
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total…
We prove that for all natural numbers $m$ and $k$ where $k$ is odd, there exists a natural number $N(k)$ such that any 3-connected cubic graph with at least $N(k)$ vertices contains a cycle of length $m$ modulo $k$. We also construct a…