Related papers: On the queue-number of graphs with bounded tree-wi…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected…

A simple graph $G$ is an {\it 2-tree} if $G=K_3$, or $G$ has a vertex $v$ of degree 2, whose neighbors are adjacent, and $G-v$ is an 2-tree. Clearly, if $G$ is an 2-tree on $n$ vertices, then $|E(G)|=2n-3$. A non-increasing sequence…

Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of…

We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for…

Let $k$ be an integer. Two vertex $k$-colorings of a graph are \emph{adjacent} if they differ on exactly one vertex. A graph is \emph{$k$-mixing} if any proper $k$-coloring can be transformed into any other through a sequence of adjacent…

We investigate relations between different width parameters of graphs, in particular balanced separator number, treewidth, and cycle rank. Our main result states that a graph with balanced separator number k has treewidth at least k but…

Treewidth is an important and well-known graph parameter that measures the complexity of a graph. The Kneser graph Kneser(n,k) is the graph with vertex set $\binom{[n]}{k}$, such that two vertices are adjacent if they are disjoint. We…

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on…

The bandwidth of a $n$-vertex graph $G$ is the smallest integer $b$ such that there exists a bijective function $f : V(G) \rightarrow \{1,...,n\}$, called a layout of $G$, such that for every edge $uv \in E(G)$, $|f(u) - f(v)| \leq b$. In…

We study the recently introduced boolean-width of graphs. Our structural results are as follows. Firstly, we show that almost surely the boolean-width of a random graph on $n$ vertices is $O(\log^2 n)$, and it is easy to find the…

A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most…

A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges.…

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the…

A connected graph has tree-depth at most $k$ if it is a subgraph of the closure of a rooted tree whose height is at most $k$. We give an algorithm which for a given $n$-vertex graph $G$, in time $\mathcal{O}(1.9602^n)$ computes the…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

A linear forest is a union of vertex-disjoint paths, and the linear arboricity of a graph $G$, denoted by $\operatorname{la}(G)$, is the minimum number of linear forests needed to partition the edge set of $G$. Clearly,…

The problem of whether and how one can compute the twin-width of a graph -- along with an accompanying contraction sequence -- lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at…

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded…