Related papers: The Zarankiewicz Problem for Polygon Visibility Gr…
A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional…
In 1975, Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked for the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$, conjecturing that $\tau = \mathcal{O}(n^{1/2})$…
We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex…
We study the problem of colouring visibility graphs of polygons. In particular, for visibility graphs of simple polygons, we provide a polynomial algorithm for 4-colouring, and prove that the 5-colourability question is already NP-complete…
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $\omega$ has chromatic number at most $3\cdot…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
We present approximation algorithms with O(n^3) processing time for the minimum vertex and edge guard problems in simple polygons. It is improved from previous O(n^4) time algorithms of Ghosh. For simple polygon, there are O(n^3) visibility…
We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite…
We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on…
We give the lower bound for the growth of the maximum value for a solution to the minimal surface equation with 0 boundary values over an unbounded simply connected domain.
For a graph $G$, its $k$-th graph power $G^k$ is constructed by placing an edge between two vertices if they are within distance $k$. We consider the problem of deriving upper bounds on the Shannon capacity of graph powers by using spectral…
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…
We study the Zarankiewicz problem for $r$-partite, $r$-uniform intersection hypergraphs arising from $r$ families of axis-parallel boxes in $\mathbb{R}^d$ with prescribed directions $F_1, \dots, F_r \subseteq \{1, \dots, d\}$. This extends…
Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox,…
A visibility representation is a classical drawing style of planar graphs. It displays the vertices of a graph as horizontal vertex-segments, and each edge is represented by a vertical edge-segment touching the segments of its end vertices;…
We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{untangling problem} is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let $fix(G)$ denote the maximum…
In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \in \{5,6 \}$. Writing $\textrm{ex}_r ( N, \mathcal{C}_{<g} )$ for this maximum, it is shown that $\textrm{ex}_r ( N ,…
We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…
In communication field, an important issue is to group users and base stations to as many as possible subnetworks satisfying certain interference constraints. These problems are usually formulated as a graph partition problems which…
We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)}…