The shortest path problem in the Knödel graph

The Knödel graph W Δ , n is a graph of even order n and degree Δ where 2 ≤ Δ ≤ ⌊ log 2 ⁡ n ⌋ . Most properties of the Knödel graph are known only for W Δ , 2 Δ and W Δ − 1 , 2 Δ − 2 . In this paper we study the shortest path problem in W Δ , n for all Δ and n. We give a tight bound on the distance b...
Ausführliche Beschreibung

The Knödel graph W Δ , n is a graph of even order n and degree Δ where 2 ≤ Δ ≤ ⌊ log 2 ⁡ n ⌋ . Most properties of the Knödel graph are known only for W Δ , 2 Δ and W Δ − 1 , 2 Δ − 2 . In this paper we study the shortest path problem in W Δ , n for all Δ and n. We give a tight bound on the distance between any two vertices in W Δ , n . We show that for almost all Δ, the presented bound differs from actual distance by at most 2. The proofs are constructive and allow us to present an O ( log ⁡ n ) time algorithm to construct a short path between any pair of vertices in W Δ , n . The diameter of W Δ , n is known only for n = 2 Δ . Using our results on the shortest path problem, we present tight upper and lower bounds on the diameter of the Knödel graph for all Δ and n. Ausführliche Beschreibung