Related papers: Chord theorems on graphs

Given a graph $G$ then a subgraph $H$ is $isometric$ if, for every pair of vertices $u,v$ of $H$, we have $d_H(u,v) = d_G(u,v)$. We say a graph $G$ is $distance\ preserving\ (dp)$ if it has an isometric subgraph of every possible order up…

Given two graphs G and H, we ask under which conditions there is a relation R that generates the edges of H given the structure of graph G. This construction can be seen as a form of multihomomorphism. It generalizes surjective…

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of…

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…

The clique vector $\mathfrak{c}(G)$ of a graph $G$ is the sequence $(c_1, c_2, \ldots,c_d)$ in $\mathbb{N}^d$, where $c_i$ is the number of cliques in $G$ with $i$ vertices and $d$ is the largest cardinality of a clique in $G$. In this…

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if…

A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $\ell \leq k$ can be connected by $\ell$ disjoint paths in the whole graph. We characterise the…

We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the…

We investigate a graph-theoretic approach to the problem of distinguishing quantum product states in the fundamental quantum communication framework called local operations and classical communication (LOCC). We identify chordality as the…

A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is…

Thomassen's chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph $G$ are the length of its longest cycles and the length of…

Let $C$ be a smooth, convex curve on either the sphere $\mathbb{S}^{2}$, the hyperbolic plane $\mathbb{H}^{2}$ or the Euclidean plane $\mathbb{E}^{2}$, with the following property: there exists $\alpha$, and parameterizations $x(t), y(t)$…

A digraph is connected-homogeneous if every isomorphism between two finite connected induced subdigraphs extends to an automorphism of the whole digraph. In this paper, we completely classify the countable connected-homogeneous digraphs.

A monomial ideal $I$ is said to have homological linear quotients if for each $k\geq 0$, the homological shift ideal $\mathrm{HS}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear…

A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end…

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. The transversal number $\tau(H)$ of $H$ is the minimum number of vertices that intersect every edge. Chv\'{a}tal and McDiarmid [Combinatorica 12 (1992),…

A curve has the increasing chord property if for any points $a,b,c,d$ in this order on the curve, the distance of $a,d$ is not smaller than that of $b,c$. Answering a conjecture of Larman and McMullen, Rote proved in 1994 that the arclength…

We generalize the notion of linear chord diagrams to the case of matched sets of size $k$, which we call $k$-chord diagrams. We provide formal generating functions and recurrence relations enumerating these $k$-chord diagrams by the number…

The purpose of this paper is twofold. We investigate a simple necessary condition, called the rhombus criterion, for two vertices in a polytope not to form an edge and show that in many examples of $0/1$-polytopes it is also sufficient. We…

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a…