An edge ordering problem of regular hypergraphs

Abstract

Given a pair of integers 2 < s < k, define gs(k) to be the minimum integer such that, for any regular multiple hypergraph H = ({1,..., k}, {e1,...,em}) with edge size at most s, there is a permutation π on {1,...,m} (or edge ordering eπ(1),....e π(m)(m)) such that g(H,π) = max{max{|dHj(u)-d Hj(v)|:u,v ∈ e(j+1)}: j = 0,...,m - 1} ≤ g s(k), where Hj = ({1,..., k}, {eπ(1),... eπ(j)}). The so-called edge ordering problem is to determine the value of gs(k) and to find a permutation π such that g(H,π) ≤ gs(k). This problem was raised from a switch box design problem, where the value of gs(k) can be used to design hyperuniversal switch boxes and an edge ordering algorithm leads to a routing algorithm. In this paper, we show that (1) g2 (k) = 1 for all k ≥ 3, (2) g s(k) = 1 for 3 ≤ s ≤ k ≤ 6, and (3) gs(k) ≤ 2k for all k ≥ 7. We give a heuristic algorithm for the edge ordering and conjecture that there is a constant C such that gs(k) ≤ C for all k and s. © Springer-Verlag Berlin Heidelberg 2006.