Related papers: Exploring Projective Norm Graphs

For integers \(r\ge 2\), \(t\ge 1\) and a real number \(a\in(3/2,2]\), we study the typical structure of oriented graphs and digraphs that do not contain a blow-up \(T_{r+1}^t\) of a transitive tournament. We prove that almost every…

Fix graphs $F$ and $H$ and let $ex(n,H,F)$ denote the maximum possible number of copies of the graph $H$ in an $n$-vertex $F$-free graph. The systematic study of this function was initiated by Alon and Shikhelman [{\it J. Comb. Theory, B}.…

A graph is said to be \textit{$H$-minor free} if it does not contain $H$ as a minor. In this paper, we characteristic the unique extremal graph with maximal $Q$-index among all $n$-vertex $K_{1,t}$-minor free graphs ($t\ge3$).

Here we consider the hypergraph Tur\'an problem in uniformly dense hypergraphs as was suggested by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density $\pi_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for…

Let $k\ge 2$ and $n_1\ge n_2\ge n_3\ge n_4$ be integers such that $n_4$ is sufficiently larger than $k$. We determine the maximum number of edges of a 4-partite graph with parts of sizes $n_1,\dots, n_4$ that does not contain $k$…

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex…

A graph that contains a spanning tree of diameter at most $t$ clearly admits a tree $t$-spanner, since a tree $t$-spanner of a graph $G$ is a sub tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times…

Spectral graph theory studies how the eigenvalues of a graph relate to the structural properties of a graph. In this paper, we solve three open problems in spectral extremal graph theory which generalize the classical Tur\'{a}n-type…

The classical K\H{o}v\'ari-S\'os-Tur\'an theorem states that if $G$ is an $n$-vertex graph with no copy of $K_{s,t}$ as a subgraph, then the number of edges in $G$ is at most $O(n^{2-1/s})$. We prove that if one forbids $K_{s,t}$ as an…

Finding dense components in graphs is of great importance in analyzing the structure of networks. Popular and computationally feasible frameworks for discovering dense subgraphs are core and truss decompositions. Recently, Sariyuce et al.…

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results…

Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any three there is an orthogonal pair. How large can $n$ be as a function of $d$, and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these two…

Pairwise Compatibility Graphs (PCGs) form a tree-metric graph class that originated in phylogeny and has since attracted sustained interest in graph theory. Several natural generalizations have been proposed in order to overcome the…

We introduce a ``Kirchhoff--Tur\'an'' variant of the extremal $C_4$ problem: among all simple connected $n$-vertex $C_4$-free graphs $G$, maximize the number of spanning trees $\tau(G)$. For the projective-plane orders $n=q^2+q+1$ we…

Let $t,q$ and $n$ be positive integers. Write $[q] = \{1,2,\ldots,q\}$. The generalized Hamming graph $H(t,q,n)$ is the graph whose vertex set is the cartesian product of $n$ copies of $[q]$ ($q\ge 2$), where two vertices are adjacent if…

Given a graph $H$, we say that a graph $G$ is $H$-saturated if $G$ contains no copy of $H$ but adding any new edge to $G$ creates a copy of $H$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices…

Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in $[n]=\{1,2,...,n\}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1\leq r\leq k\leq n$, is the graph with vertex set $[n]^{(k)}$ and with two $k$-tuples are…

For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[…

We study the behaviour of $K_{r+1}$-free graphs $G$ of almost extremal size, that is, typically, $e(G)=ex(n,K_{r+1})-O(n)$. We show that such graphs must have a large amount of 'symmetry', in particular that all but very few vertices of $G$…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…