Related papers: Forbidden cycles in metrically homogeneous graphs

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all…

For a finite graph $\Gamma$, let $G(\Gamma)$ be the right-angled Artin group defined by the complement graph of $\Gamma$. We show that, for any linear forest $\Lambda$ and any finite graph $\Gamma$, $G(\Lambda)$ can be embedded into…

Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $\Gamma(G)$ of \( G…

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$…

The well-known Erd\H{o}s-Hajnal conjecture states that for any graph $F$, there exists $\epsilon>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon}$. We consider…

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…

We use a randomised embedding method to prove that for all \alpha>0 any sufficiently large oriented graph G with minimum in-degree and out-degree \delta^+(G),\delta^-(G)\geq (3/8+\alpha)|G| contains every possible orientation of a Hamilton…

We classify the ultrahomogeneous complete 3-edge-coloured graphs (3-graphs) with simple theory. This extends Lachlan's result (a corollary of the Effective Classification Theorem for stable structures) classifying the stable homogeneous…

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ and $D\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\not=0$. We show $\Gamma$ is 3-bounded in the sense of the article [D-bounded…

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ denotes the subspace of $C(G)$, spanned by the…

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability $p \gg \frac{\log n}{n}$, the random graph $G(n,p)$ is…

In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for $k \geq 2$, a {\em non-trivial Berge $k$-cycle} is a family of sets $e_1,e_2,\dots,e_k$ such that $e_1 \cap e_2, e_2 \cap…

Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of…

We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $\Gamma(M_n(\mathbb F))$ is connected and its diameter is four.

We conjecture that every oriented graph $G$ on $n$ vertices with $\delta ^+ (G) , \delta ^- (G) \geq 5n/12$ contains the square of a Hamilton cycle. We also give a conjectural bound on the minimum semidegree which ensures a perfect packing…

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a…

Let $G$ be a finite non-cyclic group. The non-cyclic graph $\Gamma_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G:…

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…

Let $\Gamma$ be a discrete countable group acting isometrically on a measurable field $\mathbf{X}$ of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability $\Gamma$-space $(\Omega,\mu)$. If $\mathbf{X}$…

We complement the characterization of the graph products of cyclic groups $G(\Gamma, \mathfrak{p})$ admitting a Polish group topology of [9] with the following result. Let $G = G(\Gamma, \mathfrak{p})$, then the following are equivalent:…