Related papers: Subsets of Cayley graphs that induce many edges

We study some properties of the Cayley graph of the R.Thompson's group F in generators $x_0$, $x_1$. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3.…

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 $G$,…

We continue the study of the properties of graphs in which the ball of radius $r$ around each vertex induces a graph isomorphic to the ball of radius $r$ in some fixed vertex-transitive graph $F$, for various choices of $F$ and $r$. This is…

In a recent work, Keusch proved the so-called 1-2-3 Conjecture, raised by Karo\'nski, {\L}uczak, and Thomason in 2004: for every connected graph different from $K_2$, we can assign labels~$1,2,3$ to the edges so that no two adjacent…

In this paper we prove that for every $s\geq 2$ and every graph $H$ the following holds. Let $G$ be a graph with average degree $\Omega_H(s^{C|H|^2})$, for some absolute constant $C>0$, then $G$ either contains a $K_{s,s}$ or an induced…

A piece of a labelled graph $\Gamma$ defined by D. Gruber is a labelled path that embeds into $\Gamma$ in two essentially different ways. We prove that graphical $Gr'(\frac{1}{6})$ small cancellation groups whose associated pieces have…

Let $G$ be a finite abelian abelian group of exponent $m\ge 2$. For subsets $A,S\subset G$, denote by $\partial_S(A)$ the number of edges from $A$ to its complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$. We show…

For a digraph $\Gamma$, a subset $C$ of $V(\Gamma)$ is a perfect code if $C$ is a dominating set such that every vertex of $\Gamma$ is dominated by exactly one vertex in $C$. In this paper, we classify strongly connected 2-valent Cayley…

We introduce the vertex-arboricity of group-labelled graphs. For an abelian group $\Gamma$, a $\Gamma$-labelled graph is a graph whose edges are labelled by elements of $\Gamma$. For an abelian group $\Gamma$ and $A\subseteq \Gamma$, the…

We answer two extremal questions about odd cycles that naturally arise in the study of sparse pseudorandom graphs. Let $\Gamma$ be an $(n,d,\lambda)$-graph, i.e., $n$-vertex, $d$-regular graphs with all nontrivial eigenvalues in the…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and…

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph…

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma…

A long-standing and well-known conjecture (see e.g. Caro, Discrete Math, 1994) states that every $n$-vertex graph $G$ without isolated vertices contains an induced subgraph where all vertices have an odd degree and whose order is linear in…

We prove a new generalisation of Ramsey's theorem by showing that every $2$-edge-coloured graph with sufficiently large minimum degree contains a monochromatic induced subgraph whose minimum degree remains large. From this, we also derive…

Let $G$ denote a finite abelian group with identity 1 and let $S$ denote an inverse-closed subset of $G \setminus {1}$, which generates $G$ and for which there exists $s \in S$, such that $\la S \setminus \{s,s^{-1}\} \ra \ne G$. In this…

Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has maximum degree at least $\sqrt{d}$,…

We consider classes of pseudo-random graphs on $n$ vertices for which the degree of every vertex and the co-degree between every pair of vertices are in the intervals $(np - Cn^\delta,np+Cn^\delta)$ and $(np^2- C n^\delta, np^2 +C…

Given a finite Abelian group $G$ and a generator subset $A\subset G$ of cardinality two, we consider the Cayley digraph $\Gamma=$Cay$(G,A)$. This digraph is called $2$--Cayley digraph. An extension of $\Gamma$ is a $2$--Cayley digraph,…