More Results on The Smallest One-Realization of A Given Set II

• Main Authors: Diao Kefeng, Lu Fuliang, Zhao Ping
• Format: Article
• Language: English
• Published: Sciendo 2019-05-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: mixed hypergraph; feasible set; chromatic spectrum; gap; onerealization; 05c15; 05c35
• Online Access: https://doi.org/10.7151/dmgt.2093

Let S be a finite set of positive integers. A mixed hypergraph ℋ is a onerealization of S if its feasible set is S and each entry of its chromatic spectrum is either 0 or 1. The minimum number of vertices, denoted by δ3(S), in a 3-uniform bi-hypergraph which is a one-realization of S was determined in [P. Zhao, K. Diao and F. Lu, More result on the smallest one-realization of a given set, Graphs Combin. 32 (2016) 835–850]. In this paper, we consider the minimum number of edges in a 3-uniform bi-hypergraph which already has the minimum number of vertices with respect of being a minimum bihypergraph that is one-realization of S. A tight lower bound on the number of edges in a 3-uniform bi-hypergraph which is a one-realization of S with δ3(S) vertices is given.