Related papers: Packing the largest trees in the tree packing conj…

We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of…

Asymptotic behaviour of maximum sizes of induced trees and forests has been studied extensively in last decades, though the overall picture is far from being complete. In this paper, we close several significant gaps: 1) We prove $2$-point…

The graph parameter treedepth is minor-monotone; hence, the class of graphs with treedepth at most $k$ is minor-closed. By the Graph Minor Theorem, such a class is characterized by a finite set of forbidden minors. A conjecture of…

Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this paper, we give a…

This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is called \textit{$\mathcal{F}$-free} if for any $F\in \mathcal{F}$, there is no subgraph of $G$ isomorphic to $F$. Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized…

The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1984, Akiyama et al. stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum…

We propose the following conjecture: For every fixed $\alpha\in [0,\frac 13)$, each graph of minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains each tree with $k$ edges as a subgraph. Our main…

The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of…

We present a nice result on the probability of a cycle occurring in a randomly generated graph. We then provide some extensions and applications, including the proof of the famous Cayley formula, which states that the number of labeled…

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for every \epsilon>0 and positive…

Let $T$ be an oriented tree on $n$ vertices with maximum degree at most $e^{o(\sqrt{\log n})}$. If $G$ is a digraph on $n$ vertices with minimum semidegree $\delta^0(G)\geq(\frac12+o(1))n$, then $G$ contains $T$ as a spanning tree, as…

Alspach [{\sl Bull. Inst. Combin. Appl.}~{\bf 52} (2008), 7--20] defined the maximal matching sequencibility of a graph $G$, denoted~$ms(G)$, to be the largest integer $s$ for which there is an ordering of the edges of $G$ such that every…

In this paper, we focus on the prediction phase of a random forest and study the problem of representing a bag of decision trees using a smaller bag of decision trees, where we only consider binary decision problems on the binary domain and…

A $k$-ranking of a graph $G$ is a labeling of its vertices from $\{1,\ldots,k\}$ such that any nontrivial path whose endpoints have the same label contains a larger label. The least $k$ for which $G$ has a $k$-ranking is the ranking number…

The celebrated result of Koml\'os, S\'ark\"ozy, and Szemer\'edi states that for any $\varepsilon>0$, there exists $0<c<1$, such that for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\geq(1/2+\varepsilon)n$ contains…

An induced packing of cycles in a graph is a set of vertex-disjoint cycles with no edges between them. We generalise the classic Erd\H{o}s-P\'osa theorem to induced packings of cycles. More specifically, we show that there exist functions…

Fix $k \in \mathbb{N}$ and let $G$ be a connected graph with treewidth at most $k$. We say that $xy \notin E(G)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T, \cB)$ of $G$ with width at most $k$, both $x$ and $y$ are…

The famous Ryser--Brualdi--Stein conjecture asserts that every $k \times k$ Latin square contains a partial transversal of size $k-1$. Since its appearance, the conjecture has attracted significant interest, leading to several proposed…

The Erd\"os-S\'os conjecture states that if $G$ is a graph with average degree more than $k-1$, then G contains every tree of $k$ edges. A spider is a tree with at most one vertex of degree more than 2. In this paper, we prove that…