Related papers: Triangle-free geometric intersection graphs with n…
A triangle-free graph G is called k-existentially complete if for every induced k-vertex subgraph H of G, every extension of H to a (k+1)-vertex triangle-free graph can be realized by adding another vertex of G to H. Cherlin asked whether…
We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made…
The triangle-degree of a vertex v of a simple graph G is the number of triangles in G that contain v. A simple graph is triangle-distinct if all its vertices have distinct triangle-degrees. Berikkyzy et al. [Discrete Math. 347 (2024)…
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of…
It is shown that if a planar graph admits no non-constant bounded harmonic functions then the trajectories of two independent simple random walks intersect almost surely.
The intersection graph of a family of sets $\{S_{1},S_{2},\ldots,S_{n}\}$ is a graph whose vertex set is $\{S_{1},S_{2},\ldots,S_{n}\}$ and two distinct vertices are adjacent if the intersection of the corresponding sets is non-empty.…
Nielsen proved that the maximum number of maximal independent sets (MIS's) of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction a disjoint union of $k$ cliques with sizes as close to $n/k$ as…
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the…
An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We…
Let $\mathcal{H}$ be a 3-graph on $n$ vertices. The matching number $\nu(\mathcal{H})$ is defined as the maximum number of disjoint edges in $\mathcal{H}$. The generalized triangle $F_5$ is a 3-graph on the vertex set $\{a,b,c,d,e\}$ with…
We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs,…
Ajtai, Koml\'os, and Szemer\'edi proved that for sufficiently large $t$ every triangle-free graph with $n$ vertices and average degree $t$ has an independent set of size at least $\frac{n}{100t}\log{t}$. We extend this by proving that the…
For a bridgeless cubic graph $G$, $m_3(G)$ is the ratio of the maximum number of edges of $G$ covered by the union of $3$ perfect matchings to $|E(G)|$. We prove that for any $r\in [4/5, 1)$, there exist infinitely many cubic graphs $G$…
We prove that any large even Artin group is poly-free and that any even Artin group based on a triangle graph is also poly-free.
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane.…
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 give upper and lower bounds on the number of graphs of fixed degree which have a positive density of triangles. In particular, we show that there are very few such graphs, when compared to the number of graphs without this restriction.…
A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every…
It is a longstanding conjecture that every simple drawing of a complete graph on $n \geq 3$ vertices contains a crossing-free Hamiltonian cycle. We strengthen this conjecture to "there exists a crossing-free Hamiltonian path between each…
In this short note, we prove the following analog of the K\H{o}v\'ari-S\'os-Tur\'an theorem for intersection graphs of boxes. If $G$ is the intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that $G$ contains no copy of…