An Edge Ordering Problem of Regular Hypergraphs

Abstract

Given a pair of integers 2≤s ≤k, define g _s (k) to be the minimum integer such that, for any regular multiple hypergraph H =({1, ..., k}, {e ₁, ..., e _m }) with edge size at most s, there is a permutation π on {1, ..., m} (or edge ordering e _π(1), ..., e $_{\pi({\it m})}$)such that $g(H, \pi) =\max\{ \max \{|d_{H_j}(u) - d_{H_j}(v)| : u, v\in e_{\pi(j+1)}\} : j = 0, \dots, m-1\} \le g_s(k)$, where H _j = ({1, ..., k}, {e _π(1),...,e $_{\pi({\it j})}$}). The so-called edge ordering problem is to determine the value of g _s (k) and to find a permutation π such that g(H, π)≤g _s (k). This problem was raised from a switch box design problem, where the value of g _s (k) can be used to design hyper-universal switch boxes and an edge ordering algorithm leads to a routing algorithm. In this paper, we show that (1) g ₂(k) = 1 for all k≥3, (2) g _s (k) = 1 for 3≤s≤k≤6, and (3) g _s (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 g _s (k) ≤C for all k and s.