假設 G=(V,E) 是一個由頂點 V 集合與邊 E 集合所構成的圖。
r >= 2 是一個有理數。 圖 G 的一個圈 r-著色是一個將頂點集 V
映射到 [0,r) 實數區間的函數 f，使得對於所有 G 上的邊 xy，
$1<= |f(x)-f(y)|<= r-1。 我們說圖 G 是一個唯一圈 r-著色圖
假如以下成立:(i)如果圖 G 上有一個圈 r-著色函數 f, (ii) 而且
G 上的所有其他圈 r 著色函數 都可以由 f
對顏色做旋轉或鏡射而得到。 令 {cal I}(G) 是圖 G
的所有獨立集的集合。 圖 G 的一個分數 r-著色是一個將 {cal
I}(G) 映射到 [0,1] 實數區間的函數 f ，使得所有的頂點 x，
sum_{xin Iin {cal I}(G)}f(I)=1 而且 sum_{Iin {cal
I}(G)}f(I)leq r。 如果圖 G 只有唯一的一個 分數 r
著色，我們說圖 G 是一個唯一分數 $r$-著色圖。

唯一圈 r-著色圖已經在文獻上被廣泛的討論。特別針對 rgeq 2
與任給的整數 g 都存在一個唯一圈 r-著色的圖
，且此圖的最小圈長度至少為 g。 而唯一的分數 r-著色圖
是一個新的概念。 在此篇論文中，我們證明了對於任何的 rgeq 2
以及任何整數 g，都存在一個唯一分數 r-著色圖
使得其最小圈長度至少為 g。 已知對於所有的圖 G,
chi_f(G)leq chi_c(G)。 我們也證明了對於所有的有理數 rgeq
r'>2 以及任意的整數 g，都存在一個圖 G 其最小圈長度至少為
g，而且 G 是唯一圈 r-著色圖，同時 G 也是唯一分數
r'-著色圖。

This thesie discusses uniquely circular colourable and uniquely fractional
colourable graphs.
Suppose G = (V;E) is a graph and r &cedil; 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) &iexcl; f(y)j · r &iexcl; 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 &cedil; 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 &cedil; 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, &Acirc;f (G) · &Acirc;c(G). We prove that for any
rational numbers r &cedil; 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.

Contents
1 Introduction 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Circular chromatic number . . . . . . . . . . . . . . . . . . . . 7
1.3 Uniquely circular r-colourable graphs . . . . . . . . . . . . . . 10
1.4 Fractional chromatic number . . . . . . . . . . . . . . . . . . . 13
1.5 Uniquely fractional colourable graphs . . . . . . . . . . . . . . 16
1.6 Results of this thesis . . . . . . . . . . . . . . . . . . . . . . . 18
2 Uniquely colourable graphs 20
2.1 Basic properties of fractional chromatic number . . . . . . . . 20
2.2 Equivalent de‾nitions . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Uniquely fractional colourable graphs . . . . . . . . . . . . . . 25
2.4 Graphs that are simultaneously uniquely circular colourable and
uniquely fractional colourable . . . . . . . . . . . . . . . . . . 28
3 Uniquely colourable graphs of large girth 35
3.1 Probabilistic construction of uniquely k-colourable graph with
large girth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Uniquely fractional colourable graph with large girth . . . . . 38
3.3 Uniquely circular colourable and uniquely fractional colourable
graph with large girth . . . . . . . . . . . . . . . . . . . . . . 46
1
4 Further Research 53
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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