Related papers: On growth functions of ordered hypergraphs

A $k$-ranking is a vertex $k$-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest $k$ such that $G$ has a $k$-ranking. For certain graphs…

The 1-2-3 Conjecture, posed by Karo\'{n}ski, {\L}uczak and Thomason, asked whether every connected graph $G$ different from $K_2$ can be 3-edge-weighted so that every two adjacent vertices of $G$ get distinct sums of incident weights. The…

A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of…

For integers $k\ge 2$ and $N\ge 2k+1$ there are $k!2^k$ canonical orderings of the edges of the complete $k$-uniform hypergraph with vertex set $[N] = \{1,2,\dots, N\}$. These are exactly the orderings with the property that any two subsets…

For an integer $r\geqslant 3$, a hypergraph on vertex set $[n]$ is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if every two distinct edges share at most one vertex. Given a family $\mathcal{H}$ of linear…

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

The Erd\H{o}s--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erd\H{o}s--Gallai Theorem was given by Kopylov: If $G$…

An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$…

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs…

Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_{n}$ with $C$ colors containing no $k$ vertex-disjoint color…

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares and has been the focus of extensive…

A directed hypergraph is a hypergraph in which the vertex set of each hyperedge is partitioned into two disjoint parts, a head and a tail. Keszegh and P\'alv\"olgyi posed the following conjecture. Let $H$ be a directed hypergraph such that…

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a…

We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain…

We consider high-order connectivity in $k$-uniform hypergraphs defined as follows: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We describe the…

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For…

Let K be a field, [n]= {1,...,n} and H=([n],E) be a hypergraph. For an integer d >= 1 the Lovasz-Saks-Schrijver ideal (LSS-ideal) L_H^K (d) in K[y_{ij}~:~(i,j) \in [n] x [d]] is the ideal generated by the polynomials $f^{(d)}_{e}=…

A mixed hypergraph is a triple $H=(V,\mathcal{C},\mathcal{D})$, where $V$ is a set of vertices, $\mathcal{C}$ and $\mathcal{D}$ are sets of hyperedges. A vertex-coloring of $H$ is proper if $C$-edges are not totally multicolored and…

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…

Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $\Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with…