Related papers: Drawings of Complete Multipartite Graphs Up to Tri…

Let $G$ be a drawing of a graph with $n$ vertices and $e>4n$ edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma of Ajtai, Chv\'atal, Newborn,…

We propose a strengthening of the conclusion in Tur\'an's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Tur\'an theorem by weakening its…

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption…

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a…

We prove that, for sufficiently large $n$, every graph of order $n$ with minimum degree at least $0.852n$ has a fractional edge-decomposition into triangles. We do this by refining a method used by Dross to establish a bound of $0.9n$. By a…

Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum…

We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$…

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to…

In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is…

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

By a poly-line drawing of a graph G on n vertices we understand a drawing of G in the plane such that each edge is represented by a polygonal arc joining its two respective vertices. We call a turning point of a polygonal arc the bend. We…

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs,…

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$…

A graph $G = (V, E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct letters $x, y \in V$, the letters $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is…

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that…

We give criteria for deciding whether or not a triangle-free simple graph is the presentation graph of a right-angled Coxeter group that is quasiisometric to some right-angled Artin group, and, if so, producing a presentation graph for such…

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing…

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$.…

We improve upper bounds on the chromatic number proven independently in \cite{reedNote} and \cite{ingo}. Our main lemma gives a sufficient condition for two paths in graph to be completely joined. Using this, we prove that if a graph has an…