Related papers: Extremal Problems on Forest Cuts and Acyclic Neigh…

The Erd\H{o}s-S\'os conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro,…

Raspaud and Wang conjectured that every triangle-free planar graph can be vertex-partitioned into an independent set and a forest. Independently, Kawarabayashi and Thomassen also remarked that this might be true, after providing another…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

The average size of connected vertex subsets of a connected graph generalises a much-studied parameter for subtrees of trees. For trees, the possible values of this parameter are critically affected by the presence or absence of vertices of…

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an…

We prove that for every set $S$ of vertices of a directed graph $D$, the maximum number of vertices in $S$ contained in a collection of vertex-disjoint cycles in $D$ is at least the minimum size of a set of vertices that hits all cycles…

We prove that every graph with $n$ vertices and at least $5n-8$ edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$ edges. It follows…

Given a graph, the sparsest cut problem asks for a subset of vertices whose edge expansion (the normalized cut given by the subset) is minimized. In this paper, we study a generalization of this problem seeking for $ k $ disjoint subsets of…

A classical result by Erd\H{o}s, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a…

The cluster of a crossing in a graph drawing in the plane is the set of the four end-vertices of its two crossed edges. Two crossings are independent if their clusters do not intersect. In this paper, we prove that every plane graph with…

A common problem in graph colouring seeks to decompose the edge set of a given graph into few similar and simple subgraphs, under certain divisibility conditions. In 1987 Wormald conjectured that the edges of every cubic graph on $4n$…

As a variant of the much studied Tur\'an number, $ex(n,F)$, the largest number of edges that an $n$-vertex $F$-free graph may contain, we introduce the connected Tur\'an number $ex_c(n,F)$, the largest number of edges that an $n$-vertex…

We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs…

Let $G$ be a $3$-connected graph with a $3$-connected (or sufficiently small) simple minor $H$. We establish that $G$ has a forest $F$ with at least $\left\lceil(|G|-|H|+1)/2\right\rceil$ edges such that $G/e$ is $3$-connected with an…

A local subgraph of a graph is the subgraph induced by the neighborhood of a vertex. Thus a graph of order $n$ has $n$ local subgraphs. A graph $G$ is called locally nonforesty if every local subgraph of $G$ contains a cycle. Clearly, a…

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an…

We consider several problems related to packing forests in graphs. The first one is to find $k$ edge-disjoint forests in a directed graph $G$ of maximal size such that the indegree of each vertex in these forests is at most $k$. We describe…

For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of…

The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list…