Related papers: On asymmetric hypergraphs

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are…

In this paper, we address the following problem due to Frankl and F\"uredi (1984). What is the maximum number of hyperedges in an $r$-uniform hypergraph with $n$ vertices, such that every set of $r+1$ vertices contains $0$ or exactly $2$…

Let $S$ be a finite set of positive integers. A mixed hypergraph ${\cal H}$ is a one-realization of $S$ if its feasible set is $S$ and each entry of its chromatic spectrum is either 0 or 1. In [P. Zhao, K. Diao, Y. Chang and K. Wang, The…

A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number…

The aim of the present paper is to prove that the maximum number of edges in a 3-uniform hypergraph on n vertices and matching number s is max{\binom(3s+2,3), \binom(n,3) - \binom(n-s,3)} for all n,s, n >= 3s+2.

A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$…

We establish a best-possible minimum codegree condition for the existence of a perfect tiling of a $3$-uniform hypergraph $H$ with copies of the generalised triangle $T$, which is the 3-uniform hypergraph with five vertices $a, b, c, d, e$…

We show that every 1-planar graph with minimum degree at least 4 has girth at most $8$, and every 1-planar graph with minimum degree at least 3 has girth at most $198$.

An unsatisfiable formula is called minimal if it becomes satisfiable whenever any of its clauses are removed. We construct minimal unsatisfiable $k$-SAT formulas with $\Omega(n^k)$ clauses for $k \geq 3$, thereby negatively answering a…

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…

Given a coloring of the edges of the complete graph on n vertices in k colors, by considering the neighbors of an arbitrary vertex it follows that there is a monochromatic diameter two subgraph on at least 1+(n-1)/k vertices. We show that…

A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known…

Let $\mathcal{F}$ be an $r$-uniform hypergraph and $G$ be a multigraph. The hypergraph $\mathcal{F}$ is a Berge-$G$ if there is a bijection $f: E(G) \rightarrow E( \mathcal{F} )$ such that $e \subseteq f(e)$ for each $e \in E(G)$. Given a…

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We…

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism…

For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar…

We determine the minimum size of $n$-factor-critical graphs and that of $k$-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of $k$-extendable non-bipartite graphs for…

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $R^3$. We show that such a graph has at least $\frac74 |V|$ edges, and construct an infinite family of maximal knotless graphs with $|E| <…

In this paper, we compute universal minimal flows of groups of automorphisms of uncountable $\omega$-homogeneous graphs, $K_n$-free graphs, hypergraphs, partially ordered sets, and their extensions with an $\omega$-homogeneous ordering. We…

Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the…