Related papers: Tree Embeddings and Tree-Star Ramsey Numbers

One of the classical topics in graph Ramsey theory is the study of which $n$-vertex graphs have Ramsey numbers that are linear in $n$. In this paper, we consider this problem in the context of directed graphs. The oriented Ramsey number of…

The Ramsey number $r(G)$ of a graph $G$ is the minimum number $N$ such that any red-blue colouring of the edges of $K_N$ contains a monochromatic copy of $G$. Pavez-Sign\'e, Piga and Sanhueza-Matamala proved that for any function $n\leq…

Luo, Tian and Wu [Discrete Math. 345 (4) (2022) 112788] conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+w$, where $w=\max\{|X|,|Y|\}$, contains a tree…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…

Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of…

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs…

A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$…

We show that there exists a graph $G$ with $O(n)$ nodes, where any forest of $n$ nodes is a node-induced subgraph of $G$. Furthermore, for constant arboricity $k$, the result implies the existence of a graph with $O(n^k)$ nodes that…

It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page…

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the…

Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane…

The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and…

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of G that is a tree. In this paper, we study the problem of bounding t(G) for graphs which do not contain a complete graph K_r on r vertices. This problem…

We prove that any \(2\)-connected graph \(G\) on \(n\) vertices with minimum degree \(\delta(G) \ge \frac{n}{4}+2\) contains a \(2\)-connected subgraph of order \(k\) for every integer \(k\) with \(4 \le k \le n\). This improves a previous…

We consider a variation of Ramsey numbers introduced by Erd\H{o}s and Pach (1983), where instead of seeking complete or independent sets we only seek a $t$-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least…

For a graph $G$, let $a(G)$ denote the maximum size of a subset of vertices that induces a forest. We prove the following. 1. Let $G$ be a graph of order $n$, maximum degree $\Delta>0$ and maximum clique size $\omega$. Then \[ a(G) \geq…

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$,…

The bidimensionality of a set of vertices $X$ in a graph $G$ is the maximum $k$ for which $G$ contains as a $X$-rooted minor the $(k \times k)$-grid. This notion allows for the following version of the Graph Minors Structure Theorem (GMST)…

We propose the conjecture that every graph $G$ of order $n$ with less than $3n-6$ edges has a vertex cut that induces a forest. Maximal planar graphs do not have such vertex cuts and show that the density condition would be best possible.…

For integer $k\geq2,$ a graph $G$ is called $k$-leaf-connected if $|V(G)|\geq k+1$ and given any subset $S\subseteq V(G)$ with $|S|=k,$ $G$ always has a spanning tree $T$ such that $S$ is precisely the set of leaves of $T.$ Thus a graph is…