Related papers: Hypertrees and their host trees: a survey

A C-coloring of a hypergraph ${\cal H}=(X,{\cal E})$ is a vertex coloring $\varphi: X\to {\mathbb{N}}$ such that each edge $E\in{\cal E}$ has at least two vertices with a common color. The related parameter $\overline{\chi}({\cal H})$,…

Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2…

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with…

A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the…

Hypertrees are high-dimensional counterparts of graph theoretic trees. They have attracted a great deal of attention by various investigators. Here we introduce and study Hyperpaths -- a particular class of hypertrees which are high…

It is known that any chordal graph on $n$ vertices can be represented as the intersection of $n$ subtrees in a tree on $n$ nodes. This fact is recently used in [2] to generate random chordal graphs on $n$ vertices by generating $n$ subtrees…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple…

In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from $(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$ (supercritical case) for $\varepsilon>0$, with high probability the order of the largest…

A graph $G=(V,E)$ is said to be odd (or even, resp.) if $d_G(v)$ is odd (or even, resp.) for any $v\in V$. Trivially, the order of an odd graph must be even. In this paper, we show that every 4-edge connected graph of even order has a…

Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each…

Hypergraphs are a powerful abstraction for modeling high-order relations, which are ubiquitous in many fields. A hypergraph consists of nodes and hyperedges (i.e., subsets of nodes); and there have been a number of attempts to extend the…

We say that a tree $T$ is an $S$-Steiner tree if $S \subseteq V(T)$ and a hypergraph is an $S$-Steiner hypertree if it can be trimmed to an $S$-Steiner tree. We prove that it is NP-complete to decide, given a hypergraph $\mathcal{H}$ and…

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if…

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists \alpha(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{\alpha(i)}$. The Tur\'an…

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…

HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets. Multisets are composed of elements that have a multiplicity. When this multiplicity has positive integer values, it corresponds to non…

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex of G such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at mots s,…