Related papers: Tree-alpha and excluding finitely many graphs

We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more…

In 1998, Broersma and Tuinstra [J. Graph Theory \textbf{29} (1998), 227-237] proved that if $G$ is a connected graph satisfying $\sigma_2(G) \geq |G|-k+1$ then $G$ has a spanning $k-$ended tree. They also gave an example to show that the…

We introduce the notion of \emph{bounded diameter arboricity}. Specifically, the \emph{diameter-$d$ arboricity} of a graph is the minimum number $k$ such that the edges of the graph can be partitioned into $k$ forests each of whose…

We study a conjecture due to Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht stating that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right],$ define the matrix $A_{\alpha}\left(G\right) $ as \[ A_{\alpha}\left(G\right) =\alpha…

Two sets $X, Y$ of vertices in a graph $G$ are "anticomplete" if $X\cap Y=\varnothing$ and there is no edge in $G$ with an end in $X$ and an end in $Y$. We prove that every graph $G$ of sufficiently large treewidth contains two anticomplete…

We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the…

In the Fully Leafed Induced Subtrees, one is given a graph $G$ and two integers $a$ and $b$ and the question is to find an induced subtree of $G$ with $a$ vertices and at least $b$ leaves. This problem is known to be NP-complete even when…

The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about…

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in…

We show that every connected graph can be approximated by a normal tree, up to some arbitrarily small error phrased in terms of neighbourhoods around its ends. The existence of such approximate normal trees has consequences of both…

In this paper, we establish that the class of $\{P_6, (2,2)\text{-broom}\}$-free graphs contains a subclass $\mathcal{L}_i$, defined by certain cutset conditions, whose chromatic number admits a linear $\chi$-bound. Building on recent…

We define the induced arboricity of a graph $G$, denoted by ${\rm ia}(G)$, as the smallest $k$ such that the edges of $G$ can be covered with $k$ induced forests in $G$. This notion generalizes the classical notions of the arboricity and…

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…

Given a multigraph $G$ whose edges are colored from the set $[q]:=\{1,2,\ldots,q\}$ (\emph{$q$-colored graph}), and a vector $\alpha=(\alpha_1,\ldots,\alpha_{q}) \in \mathbb{N}^{q}$ (\emph{color-constraint}), a subgraph $H$ of $G$ is called…

Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that…

Let $G$ be a graph and $\mathcal{F}$ a family of graphs. Define $\alpha_{\mathcal{F}}(G)$ as the maximum order of any induced subgraph of $G$ that belongs to the family $\mathcal{F}$. For the family $\mathcal{F}$ of graphs with…

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,…

A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H…

We prove that for every graph H and for every integer s, the class of graphs that do not contain K_s, K_{s,s}, or any subdivision of H as an induced subgraph has bounded expansion; this strengthens a result of Kuhn and Osthus. The argument…