Related papers: Conditional Information Inequalities and Combinato…

The Ingleton inequality is a classical linear information inequality that holds for representable matroids but fails to be universally valid for entropic vectors. Understanding the extent to which this inequality can be violated has been a…

Let $G=(S,T,E)$ be a bipartite graph. For a matching $M$ of $G$, let $V(M)$ be the set of vertices covered by $M$, and let $B(M)$ be the symmetric difference of $V(M)$ and $S$. We prove that if $M$ is a uniform random matching of $G$, then…

An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as…

A graph $G = (V, E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct letters $x, y \in V$, the letters $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is…

Haxell's condition is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the…

A folklore result on matchings in graphs states that if $G$ is a bipartite graph whose vertex classes $A$ and $B$ each have size $n$, with $\mathrm{deg}(u) \geq a$ for every $u \in A$ and $\mathrm{deg}(v) \geq b$ for every $v \in B$, then…

In this paper we consider finite conditional random quantities and conditional previsions assessments in the setting of coherence. We use a suitable representation for conditional random quantities; in particular the indicator of a…

Cohen-Macaulayness of bipartite graphs is investigated by several mathematicians and has been characterized combinatorially. In this note, we give some different combinatorial conditions for a bipartite graph which are equal to…

If $G$ is a bipartite graph, Hall's theorem \cite{H35} gives a condition for the existence of a matching of $G$ covering one side of the bipartition. This theorem admits a well-known algorithmic proof involving the repeated search of…

We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information…

For a bivariate random vector (X,Y), symmetry conditions are presented that yield stochastic orderings among |X|, |Y|, |max(X,Y)|, and | min(X, Y)|. Partial extensions of these results for multivariate random vectors (X1,...,Xn) are also…

The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in…

The conditional mutual information I(X;Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model…

Let $G$ be a uniformly chosen simple (labelled) random graph with given degree sequence $\boldsymbol{d}$ and let $X,Y,L$ be edge-disjoint graphs on the same vertex set as $G$. We investigate the probability that $X \subseteq G$ and that $G…

A bipartite graph $H$ is said to have Sidorenko's property if the probability that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$ is a homomorphism is at least the product over all edges in $H$ of the probability…

In this paper, we give sufficient conditions on the spectral radius for a bipartite graph to Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also present tight sufficient conditions on…

The bipartite-hole-number of a graph $G$, denoted as $\widetilde{\alpha}(G)$, is the minimum number $k$ such that there exist positive integers $s$ and $t$ with $s+t=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$…

The celebrated dependent random choice lemma states that in a bipartite graph an average vertex (weighted by its degree) has the property that almost all small subsets $S$ in its neighborhood has common neighborhood almost as large as in…