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Optimal orientations of the torus
Author
Chew, Kah Yee
Supervisor
Tay, Eng Guan
Abstract
For a graph G, let ₣ ( G ) be the family of strong orientations of G. We define the oriented diameter of G as D(G) = min {D(F) I F E ₣ ( G ) ) . Any orientation F in ₣ ( G )with D(F) = D (G) is called an optimal orientation of G. In this academic exercise, we study the optimal orientations of the torus, which is the cartesian product of two cycles.
Our main objective is to provide the opt,imal orientation for a CH X CM torus when at least one of M and N is odd. Of particular interest is the optimal oriented diameter of the Cdn+3 X C5 torus which was listed as a conjecture by Konig et al. [l]. We prove that the torus has an optimal oriented diameter of D + 1. Other proofs included are for cases where:
● N = 2 (mod4), M ≥ 5 and M odd,
● N = 0 (mod4), M = 5,
● N = 3, M = 3,
● N ≥ 7, M ≥ 7, where N, M are odd,
When M is even ;we also briefly explain the need for providing different orientations for the case where N = 2 (mod4) and the case where N = 0 (mod4).
Our main objective is to provide the opt,imal orientation for a CH X CM torus when at least one of M and N is odd. Of particular interest is the optimal oriented diameter of the Cdn+3 X C5 torus which was listed as a conjecture by Konig et al. [l]. We prove that the torus has an optimal oriented diameter of D + 1. Other proofs included are for cases where:
● N = 2 (mod4), M ≥ 5 and M odd,
● N = 0 (mod4), M = 5,
● N = 3, M = 3,
● N ≥ 7, M ≥ 7, where N, M are odd,
When M is even ;we also briefly explain the need for providing different orientations for the case where N = 2 (mod4) and the case where N = 0 (mod4).
Date Issued
2004
Call Number
QA166 Che
Date Submitted
2004