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The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe…
We examine the dependence structure of finite block-maxima of multivariate distributions. We provide a closed form expression for the copula density of the vector of the block-maxima. Further, we show how partial derivatives of…
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
Three science and engineering problems of recent interests -index coding, locally recoverable distributed storage, and guessing games on graphs- are discussed and the connection between their optimal solutions is elucidated. By generalizing…
The number of triangles in a graph is useful to deduce a plethora of important features of the network that the graph is modeling. However, finding the exact value of this number is computationally expensive. Hence, a number of…
We consider the capacity problem for wireless networks. Networks are modeled as random unit-disk graphs, and the capacity problem is formulated as one of finding the maximum value of a multicommodity flow. In this paper, we develop a proof…
Social networks have become an inseparable part of human life and processing them in an efficient manner is a top priority in the study of networks. These networks are highly dynamic and they are growing incessantly. Inspired by the concept…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
A casino offers the following game. There are three cups each containing a die. You are being told that the dice in the cups are all the same, but possibly nonstandard. For a bet of \$1, the game master shakes all three cups and lets you…
A famous conjecture of Tuza \cite{tuza} is that the minimal number of edges needed to cover all triangles in a graph is at most twice the maximal number of edge-disjoint triangles. We propose a wider setting for this conjecture. For a…
In the graph sharing game, two players share a connected graph $G$ with non-negative weights assigned to the vertices, claiming and collecting the vertices of $G$ one by one, while keeping the set of all claimed vertices connected through…
Graphs are used to model interactions in a variety of contexts, and there is a growing need to quickly assess the structure of such graphs. Some of the most useful graph metrics are based on triangles, such as those measuring social…
We consider the problem of estimating the topology of multiple networks from nodal observations, where these networks are assumed to be drawn from the same (unknown) random graph model. We adopt a graphon as our random graph model, which is…
Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a…
We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erd\H{o}s' Sparse Half Conjecture. Among others we prove the following variant of it: For every sufficiently large even integer $n$ the…
The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide…
We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set…
In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically…
We consider three variants of the problem of finding a maximum weight restricted $2$-matching in a subcubic graph $G$. (A $2$-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on…
The well-known 1-2-3 Conjecture asserts that the edges of every graph without isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices receive distinct weighted degrees. This is open in general, while it is known to be…