Related papers: Skewness, crossing number and Euler's bound for gr…

The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a…

The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of…

We provide a new lower bound on the number of $(\leq k)$-edges of a set of $n$ points in the plane in general position. We show that for $0 \leq k \leq\lfloor\frac{n-2}{2}\rfloor$ the number of $(\leq k)$-edges is at least $$ E_k(S) \geq…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving…

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, Masa\v{r}\'ik and Hlin\v{e}n\'y [GD 2023] recently asked for each edge of $G$ to be drawn without…

We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

A {\it universal labeling} of a graph $G$ is a labeling of the edge set in $G$ such that in every orientation $\ell$ of $G$ for every two adjacent vertices $v$ and $u$, the sum of incoming edges of $v$ and $u$ in the oriented graph are…

The crossing number of a graph $G$ is the minimum number of crossings in a drawing of $G$ in the plane. A rectilinear drawing of a graph $G$ represents vertices of $G$ by a set of points in the plane and represents each edge of $G$ by a…

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper…

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound.…

The imbalance of an edge $e=\{u,v\}$ in a graph is defined as $i(e)=|d(u)-d(v)|$, where $d(\cdot)$ is the vertex degree. The irregularity $I(G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$. This concept was…

The classical Crossing Lemma by Ajtai et al.~and Leighton from 1982 gave an important lower bound of $c \frac{m^3}{n^2}$ for the number of crossings in any drawing of a given graph of $n$ vertices and $m$ edges. The original value was $c=…

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the…

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by [Thomassen '89 & '93], and it is known to be fixed-parameter tractable. However, the…

The {\it toughness} $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chv\'{a}tal in 1973. A graph $G$ is called {\it $t$-tough} if $\tau(G)\geq t.$ Let…

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. In this paper, we give the exact values of crossing numbers for some variations of hypercube with order at most four,…

A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a…

Dujmovi\'c, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every graph $G$ with Euler genus $g$ there is a graph $H$ with treewidth at most 4 and a path $P$ such that $G\subseteq H \boxtimes P \boxtimes…