Related papers: Local conditions for exponentially many subdivisio…

After seeing how questions on the finer distribution of prime factorization -- considered inaccessible until recently -- reduce to bounding the norm of an operator defined on a graph describing factorization, we will show how to bound that…

A widely open conjecture proposed by Bollob\'as, Erd\H{o}s, and Tuza in the early 1990s states that for any $n$-vertex graph $G$, if the independence number $\alpha(G) = \Omega(n)$, then there is a subset $T \subseteq V(G)$ with $|T| =…

Let $G$ be a simple and finite graph without isolated vertices. In this paper we study forcing sets (zero forcing sets) which induce a subgraph of $G$ without isolated vertices. Such a set is called a total forcing set, introduced and first…

A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by the union of S and N(S), where N(S) is the neighborhood of S. A matching M is…

A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected…

In 2009, Krivelevich and Sudakov studied the existence of large complete minors in $(t,\alpha)$-expanding graphs whenever the expansion factor $t$ becomes super-constant. In this paper, we give an extension of the results of Krivelevich and…

This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy…

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $\iota(G,F)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$…

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and,…

For every integer $t \ge 0$, denote by $F_5^t$ the hypergraph on vertex set $\{1,2,\ldots, 5+t\}$ with hyperedges $\{123,124\} \cup \{34k : 5 \le k \le 5+t\}$. We determine $\mathrm{ex}(n,F_5^t)$ for every $t\ge 0$ and sufficiently large…

The $t$-vertex condition, for an integer $t\ge 2$, was introduced by Hestenes and Higman in 1971, providing a combinatorial invariant defined on edges and non-edges of a graph. Finite rank 3 graphs satisfy the condition for all values of…

A graph is locally irregular if any pair of adjacent vertices have distinct degrees. A locally irregular decomposition of a graph $G$ is a decomposition $\mathcal{D}$ of $G$ such that every subgraph $H \in \mathcal{D}$ is locally irregular.…

For graphs F and G an F-matching in G is a subgraph of G consisting of pairwise vertex disjoint copies of F. The number of F-matchings in G is denoted by s(F,G). We show that for every fixed positive integer m and every fixed tree F, the…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

A graph $G$ contains a strong parity factor $F$ if for every subset $X\subseteq V(G)$ with $|X|$ even, $G$ has a spanning subgraph $F$ satisfying $\delta(F)\geq1$, $d_F(u)\equiv1$ (mod 2) for any $u\in X$, and $d_F(v)\equiv0$ (mod 2) for…

Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\in H(v){4em}\hbox{for every}v\in V(G).$$ Let…

An immersion of a graph $H$ into a graph $G$ is a one-to-one mapping $f:V(H) \to V(G)$ and a collection of edge-disjoint paths in $G$, one for each edge of $H$, such that the path $P_{uv}$ corresponding to edge $uv$ has endpoints $f(u)$ and…

We consider extension variants of the classical graph problems Vertex Cover and Independent Set. Given a graph $G=(V,E)$ and a vertex set $U \subseteq V$, it is asked if there exists a minimal vertex cover (resp.\ maximal independent set)…

For a complex polynomial or analytic function f, one has been studying intensively its so-called local zeta functions or complex powers; these are integrals of |f|^{2s}w considered as functions in s, where the w are differential forms with…

We propose local versions of Hadwiger's Conjecture, where only balls of radius $\Omega(\log(v(G)))$ around each vertex are required to be $K_{t}$-minor-free. We ask: if a graph is locally-$K_{t}$-minor-free, is it $t$-colourable? We show…