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Optimal orientations of G vertex-multiplications with some techniques from Sperner Theory
Author
Wong, Willie Han Wah
Supervisor
Tay, Eng Guan
Dong, F. M.
Abstract
This thesis is concerned with the study of optimal orientations of a family of graphs, known as the G vertex-multiplications, using techniques in Sperner Theory. An optimal orientation of a bridgeless graph is a strong orientation that minimises the increase from the original diameter while the orientation number is the minimum oriented diameter. Koh and Tay [48] were the first to study the orientation numbers of G vertex multiplications,
which is an extension of complete n-partite graphs. They proved a fundamental classi cation into three classes C0; C1 and C2 (Theorem 2.2.1). Gutin characterised complete bipartite graphs (Theorem 2.1.4) with orientation number 3 (or 4 resp.) via an ingenious use of a fundamental result in Sperner Theory, namely Sperner's theorem (Theorem 2.3.1). The central theme of this thesis is to extend Gutin's approach by transforming some aspects of optimal orientations of G vertex-multiplications to problems in Sperner Theory, particularly concerning cross-intersecting Sperner families (or antichains).
Date Issued
2022
Call Number
QA166 Won
Date Submitted
2022