A Conditional Information Inequality and Its Combinatorial Applications

Abstract

We show that the inequality H(A|B, X) + H(A|B, Y ) H(A|B) for jointly distributed random variables A, B, X, Y , which does not hold in general case, holds under some natural condition on the support of the probability distribution of A, B, X, Y . This result generalizes a version of the conditional Ingleton inequality: if for some distribution I(X :We present two applications of our result. The first one is the following easy-to-formulate theorem on edge colorings of bipartite graphs: assume that the edges of a bipartite graph are colored in K colors so that each two edges sharing a vertex have different colors and for each pair (left vertex x, right vertex y) there is at most one color a such both x and y are incident to edges with color a; assume further that the degree of each left vertex is at least L and the degree of each right vertex is at least R. Then K LR. The second application is a new method to prove lower bounds for biclique cover of bipartite graphs. KeywordsShannon entropy, conditional information inequalities, non Shannon type information inequalities, biclique cover, edge coloring

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

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An Operational Characterization of Mutual Information in Algorithmic Information Theory

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Entropy-Based Proofs of Combinatorial Results on Bipartite Graphs

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