Trinque problem: covering complete graphs by plane degree-bounded hypergraphs

Abstract

Let $K_n$ be a complete graph drawn on the plane with every vertex incident to the infinite face. For any integers i and d, we define the (i, d)-Trinque Number of $K_n$, denoted by ${\mathcal {T}}^d_{i}(K_n)$, as the smallest integer k such that there is an edge-covering of $K_n$ by k “plane” hypergraphs of degree at most d and size of edge bounded by i. We compute this number for graphs (that is $i=2$) and gives some bounds for general hypergraphs.