Abstract |
This thesie discusses uniquely circular colourable and uniquely fractional colourable graphs. Suppose G = (V;E) is a graph and r ¸ 2 is a real number. A circular r-colouring of G is a mapping f : V (G) ! [0; r) such that for any edge xy of G, 1 · jf(x) ¡ f(y)j · r ¡ 1. We say G is uniquely circular r-colourable if there is a circular r-colouring f of G and any other circular r-colouring of 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 G is a mapping f : I(G) ! [0; 1] such that for any vertex x, Px2I f(I) = 1 and PI2I(G) f(I) · r. A graph G is called uniquely fractional r-colourable if there is exactly one fractional r-colouring of G. Uniquely circular r-colourable graphs have been studied extensively in the literature. In particular, it is known 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 graphs is a new concept. In this thesis, we prove that for any r ¸ 2 for any integer g, there is a uniquely fractional r-colourable graph of girth at least g. It is well-known that for any graph G, Âf (G) · Âc(G). We prove that for any rational numbers r ¸ r0 > 2 and any integer g, there is a graph G of girth at least g, which is uniquely circular r-colourable and at the same time uniquely fractional r0-colourable. |