Related papers: Minimum multiplicities of subgraphs and Hamiltonia…

A graph $G$ is said to be ubiquitous, if every graph $\Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every…

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge…

We give sufficient conditions for stability of a continuous-time linear switched system consisting of finitely many subsystems. The switching between subsystems is governed by an underlying graph. The results are applicable to switched…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

We are concerned with split graphs and pseudo-split graphs whose complements are isomorphic to themselves. These special subclasses of self-complementary graphs are actually the core of self-complementary graphs. Indeed, we show that all…

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each of its component is a star. Clearly, every graph without isolated vertices has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors of…

In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…

We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by…

Let $H$ be a fixed graph. A graph $G$ is called {\it $H$-saturated} if $H$ is not a subgraph of $G$ but the addition of any missing edge to $G$ results in an $H$-subgraph. The {\it saturation number} of $H$, denoted $sat(n,H)$, is the…

It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected…

Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…

A graph G is distinguished if its vertices are labelled by a map \phi: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves \phi. The distinguishing number of G is the minimum number k necessary for \phi to distinguish…

In this exposition, we show that a Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are smooth enough. We also provide some counterexamples…

Given two graphs G and H its 1-{\it join} is the graph obtained by taking the disjoint union of G and H and adding all the edges between a nonempty subset of vertices of G and a nonempty subset of vertices of H. In general, composition…

A graph $G$ is Hamiltonian-connected if there exists a Hamiltonian path between any two vertices of $G$. It is known that if $G$ is 2-connected then the graph $G^2$ is Hamiltonian-connected. In this paper we prove that the square of every…

A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pairs of end vertices. We give a characterization of the graphs…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…