Entropy splitting hypergraphs (Q1924130)

Let \(F\) be a hypergraph on the vertex set \(V(F)=\{1, \dots, n\}\) and let \(P=(p_1, \dots, p_n)\) be a probability distribution on \(V(F)\), i.e., \(p_1 + \cdots +p_n =1 \) and \( p_i \geq 0\) for all \(i\). The entropy of \(F\) with respect to \(P\) is definde by \[ H(F,P) = \min_{a \in VP(F)} - \sum_{i=1}^n p_i \log a_i. \] A \(k\)-uniform hypergraph \(F\) is said to be strongly splitting if for every probability distribution \(P\) on \(V(F)\), it holds \(H(F,P)+ H(\overline {F},P)=H(K_n^{(k)}, P) \), where \(K_n^{(k)} \) is the complete \(k\)-uniform hypergraph on \(n\) vertices. Let \(T\) be a two-colored tree with two colors, 0 and 1. The leaf-pattern of \(T\) is the 3-uniform hypergraph \(F\) such that vertices are the leaves of \(T\) and three leaves form an edge if and only if the unique common point of the path joining pairs of them is colored 1. The main result of the paper is the following: A 3-uniform hypergrah is strongly splitting if and only if it is a leaf-pattern of some two-colored tree \(T\).