Related papers: A counterexample to a conjecture of Ghosh

For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete…

Trees without vertices of degree $2$ are sometimes named topological trees. In this work, we bring forward the study of the inducibility of (rooted) topological trees with a given number of leaves. The inducibility of a topological tree $S$…

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose…

An ancestry labeling scheme assigns labels (bit strings) to the nodes of rooted trees such that ancestry queries between any two nodes in a tree can be answered merely by looking at their corresponding labels. The quality of an ancestry…

We generalize Schwenk's result that almost all trees contain any given limb to trees with positive integer vertex weights. The concept of characteristic polynomial is extended to such weighted trees and we prove that the proportion of…

Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be…

In graph theory, a Snark is a connected, bridgeless, Cubic graph that cannot be edge-colored with only three colors. Additionally, to avoid some trivial cases, a Snark is typically required to have a girth of minimum five and a cyclic…

As was shown recently, non-Abelian vortex strings supported in four-dimensional ${\cal N}=2$ supersymmetric QCD with the U(2) gauge group and $N_f=4$ quark multiplets (flavors) become critical superstrings. In addition to the translational…

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,…

This paper studies constructive heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree that uses edges that are as similar as possible. Given an undirected labeled connected graph (i.e.,…

We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs…

We show that the expected size of the maximum agreement subtree of two $n$-leaf trees, uniformly random among all trees with the shape, is $\Theta(\sqrt{n})$. To derive the lower bound, we prove a global structural result on a decomposition…

An $n$-vertex graph is degree 3-critical if it has $2n - 2$ edges and no proper induced subgraph with minimum degree at least 3. In 1988, Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp asked whether one can always find cycles of all short…

Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, as an extension of complete $n$-partite graphs. They proved a fundamental classification of $G$ vertex-multiplications into three classes $\mathscr{C}_0,…

We show that if $D$ is an $n$-vertex digraph with more than $(k-1)n$ arcs that does not contain any of three forbidden digraphs, then $D$ contains every antidirected tree on $k$ arcs. The forbidden digraphs are those orientations of $K_{2,…

The derived functors $\lim^n$ of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors,…

For any non-principal polarisation $D$, we explicitly construct $D$-polarised abelian variety $A$, such that its dual abelian variety is not (abstractly) isomorphic to $A$. For $\dim(A)>3$ the construction includes examples with submaximal…

A bridgeless cubic graph $G$ is said to have a 2-bisection if there exists a 2-vertex-colouring of $G$ (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an…

We study restricted computation models related to the Tree Evaluation Problem}. The TEP was introduced in earlier work as a simple candidate for the (*very*) long term goal of separating L and LogDCFL. The input to the problem is a rooted,…

A graph is chordal if it contains no induced cycle of length four or more. While finite chordal graphs are precisely those admitting tree-decompositions into cliques, this fails for infinite graphs. We establish two results extending the…