Related papers: Small intersection numbers in the curve graph

For a graph $G$ and a positive integer $k$, a vertex labelling $f:V(G)\to\{1,2\ldots,k\}$ is said to be $k$-distinguishing if no non-trivial automorphism of $G$ preserves the sets $f^{-1}(i)$ for each $i\in\{1,\ldots,k\}$. The…

A maximal geodesic in a graph is a geodesic (alias shortest path) which is not a subpath of a longer geodesic. The geodesic-transversal problem in a graph $G$ is introduced as the task to find a smallest set $S$ of vertices of $G$ such that…

Let $(\Sigma, g)$ be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics $\gamma_{\star,1}, \dots \gamma_{\star, r}$. We give an asymptotic growth as $L \to +\infty$ of the number of primitive…

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to…

We present two results related to an edge-isoperimetric question for Cayley graphs on the integer lattice asked by Ben Barber and Joshua Erde [Isoperimetry of Integer Lattices, Discrete Analysis 7 (2018)]. For any (undirected) graph $G$,…

An oriented graph is a directed graph with no bi-directed edges, i.e. if $xy$ is an edge then $yx$ is not an edge. The oriented size Ramsey number of an oriented graph $H$, denoted by $r(H)$, is the minimum $m$ for which there exists an…

Given a smooth, irreducible, projective surface $S$, let $g(S)$ be the minimum geometric genus of an irreducible curve that moves in a linear system of positive dimension on $S$. We determine the value of this birational invariant for a…

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

Let $G$ be a connected graph on $n$ vertices. The Gallai number $Gal(G)$ of $G$ is the size of the smallest set of vertices that meets every maximum path in $G$. Gr\"unbaum constructed a graph $G$ with $Gal(G)=3$. Very recently, Long,…

We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely…

Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $\kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted…

Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space…

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain…

We prove three results conjectured or stated by Chartrand, Fink and Zhang [European J. Combin {\bf 21} (2000) 181--189, Disc. Appl. Math. {\bf 116} (2002) 115--126, and pre-print of ``The hull number of an oriented graph'']. For a digraph…

Supersingular isogeny graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for the number of loops and multi-edges in such graphs. We also find conditions under which the…

We prove that the minimal length of a closed geodesic with self-intersection number $k$ on any finite-type hyperbolic surface is $2\cosh^{-1}(1+2k)$ for $k>1750$. This improves the previously known threshold $k > 10^{13350}$. Our proof is…

The strong chromatic number, $\chi_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph…

Given a graph $G$, denote by $h(G)$ the smallest size of a subset of $V(G)$ which intersects every maximum independent set of $G$. We prove that any graph $G$ without induced matching of size $t$ satisfies $h(G)\le \omega(G)^{3t-3+o(1)}$.…

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal…