Related papers: Testing Equality in Communication Graphs

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the…

An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a…

We show that every proper minor-closed class of graphs admits a $(1+o(1))\log_2 n$-bit adjacency labelling scheme. Equivalently, for every proper minor-closed class $\mathcal{G}$ and every positive integer $n$ there exists an…

Let $h(n)$ be the minimum integer such that every complete $n$-vertex simple topological graph contains an edge that crosses at most $h(n)$ other edges. In 2009, Kyn\v{c}l and Valtr showed that $h(n) = O(n^2/\log^{1/4} n)$, and in the other…

A connected graph, on four or more vertices, is matching covered (aka 1-extendable) if every edge is present in some perfect matching. An ear decomposition theorem exists for bipartite matching covered graphs due to Hetyei. From the results…

We consider graphs without loops or parallel edges in which every edge is assigned + or -. Such a signed graph is balanced if its vertex set can be partitioned into parts $V_1$ and $V_2$ such that all edges between vertices in the same part…

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer…

A line-broadcasting model in a connected graph $G=(V,E)$, $|V|=n$, is a model in which one vertex, called the {\it originator} of the broadcast holds a message that has to be transmitted to all vertices of the graph through placement of a…

A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…

Given a connected graph, the principal eigenvector of the adjacency matrix (often called the Perron vector) can be used to assign positive weights to the vertices. A natural way to measure the homogeneousness of this vector is by…

Mader [J. Combin. Theory Ser. B 40 (1986) 152-158] proved that every $k$-edge-connected graph $G$ with minimum degree at least $k+1$ contains a vertex $u$ such that $G-\{u\}$ is still $k$-edge-connected. In this paper, we prove that every…

The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the…

We study a distributed consensus problem on a complete communication network of $n$ vertices, each holding one of two opinions. The vertices communicate in rounds, possibly in the presence of adversarial noise, and exchange information…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of…

The paper presents an algorithm for minimum vertex cover problem, which is an NP-Complete problem. The algorithm computes a minimum vertex cover of each input simple graph. Tested by the attached MATLAB programs, Stage 1 of the algorithm is…

Consider the graph that has as vertices all bitstrings of length $2n+1$ with exactly $n$ or $n+1$ entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts…

We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to…