Conditional Information Inequalities and Combinatorial Applications

We provide a new relaxation of conditions under which the inequality H(A|B,X) +H(A|B,Y ) 6 H(A|B) holds for jointly distributed random variables A,B,X, Y . Then we present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: Assume that edges of a bipartite graph are partitioned into K matchings so that for each pair (left vertex x, right vertex y) there is at most one matching in the partition that involves both x, y. 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 coverings of bipartite graphs. We also provide a new relaxation of the conditions for the inequality proven by Z. Zhang, R.W. Yeung (1997).