### Title page for etd-0125106-114359

URN etd-0125106-114359 Shu-yuan Lin No Public. This thesis had been viewed 5224 times. Download 2453 times. Applied Mathematics 2005 1 Ph.D. English Simultaneously Uniquely Circular Colourable and Uniquely Fractional Colourable Graphs 2006-01-19 65 uniquely circular colourable uiquely fractional colourable Kneser graph This thesie discusses uniquely circular colourable and uniquely fractionalcolourable graphs.Suppose G = (V;E) is a graph and r ¸ 2 is a real number. A circularr-colouring of G is a mapping f : V (G) ! [0; r) such that for any edge xyof G, 1 · jf(x) ¡ f(y)j · r ¡ 1. We say G is uniquely circular r-colourableif there is a circular r-colouring f of G and any other circular r-colouringof G can obtained from f by a rotation or a °ip of the colours. Let I(G)denote the family of independent sets of G. A fractional r-colouring of Gis a mapping f : I(G) ! [0; 1] such that for any vertex x, Px2I f(I) = 1and PI2I(G) f(I) · r. A graph G is called uniquely fractional r-colourable ifthere is exactly one fractional r-colouring of G. Uniquely circular r-colourablegraphs have been studied extensively in the literature. In particular, it isknown that for any r ¸ 2, for any integer g, there is a uniquely circular r-colourable graph of girth at least g. Uniquely fractional r-colouring of graphsis a new concept. In this thesis, we prove that for any r ¸ 2 for any integerg, there is a uniquely fractional r-colourable graph of girth at least g. It iswell-known that for any graph G, Âf (G) · Âc(G). We prove that for anyrational numbers r ¸ r0 > 2 and any integer g, there is a graph G of girth atleast g, which is uniquely circular r-colourable and at the same time uniquelyfractional r0-colourable. Gerard Jenn-hwa Chang - chair D. J. Guan - co-chair Sen-Peng Eu - co-chair Yeong-Nan Yeh - co-chair Hong-Gwa Yeh - co-chair Li-Da Dong - co-chair Xu-ding Zhu - advisor indicate access worldwide 2006-01-25

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