摘 要

本論文構造具有給定的圓色數(circular chromatic number)或圓流數(circular flow number)的特殊圖。
假定$G=(V,E)$是一個圖而且$rgeq 2$是一個實數。圖$G$的一個$r$著色( )是一個映射$f:V->[0,r]$使得對任意兩個有邊相連的頂點 ,y滿足$1 |f(x)-f(y)| r-1$。一個圖G的圓色數$chi_c(G)$是指最小的$r$使得$G$存在一個$r$著色。圓色數是由Vince [33]在1988年定義，在當時是叫做星著色數(star chromatic number)記做$chi*(G)$。Vince證明了對任意的有理數$k/d>2$，存在一個圖$G$，其圓色數$chi_c(G)=k/d$。在本篇論文中，我們討論具有給定的圓色數的特殊圖的存在性。
如果一個圖$H$可以由圖$G$去掉一些點或一些邊或是收縮一些邊得到，則$H$稱為圖$G$的一個minor。如果$H$不是$G$的minor則稱$G$是$H$-minor free。著名的Hadwiger的猜想斷言對任意的正整數$n$，所有的$K_n$-minor free圖都可以$n-1$著色($(n-1)$-colorable)。如果這個猜想是正確的，則對任意的$K_n$-minor free圖$G$，$chi_c(G)=r$。另一方面，對任意至少有一條邊的圖$G$，$chi_c(G)geq2$。一個自然的問題：是否對任意的有理數$2leq rleq n-1$，存在一個$K_n$-minor free圖$G$，$chi_c(G)=r$?
當$n=4$，答案是否定的。Hell和Zhu [9] 證明如果圖$G$是$K_4$-minor free則$chi_c(G)=3$ 或$chi_c(G)leq 8/3$。所以不存在任何$K_4$-minor free圖圓色數在$(8/3,3)的有理數。當$ngeq 5$，Zhu [49] 證明對任意的有理數$rin[2,n-2]，存在一的$chi_c(G)=r$的$K_n$-minor free圖 。對任意的有理數$rin(n-2,n-1)$，是否存在$chi_c(G)=r$的$K_n$-minor free圖 的問題在本文之前尚未獲得解決。本篇論文給予一個肯定的答案。當整數ngeq 5，對任意的有理數$rin [n-2,n-1]$，我們構造一個的$K_n$-minor free圖 ,$chi_c(G)=r$。這表示當整數$ngeq5$時，對任意的有理數 $rin(n-2,n-1)$，存在一個$chi_c(G)=r$的$K_n$-minor free圖 。在$n=5$的時候，在本論文中所構造的$K_n$-minor free圖是平面圖(planar graph)。 所以我們的結果表示對任意的有理數 ，存在一個$chi_c(G)=r$的平面圖G。這個結果首先由Moser [18] 和Zhu [46] 證明。Moser證明對任意的有理數$rin[2,3]$，存在一個$chi_c(G)=r$的平面圖 ，Zhu證明對任意的有理數$rin[3,4]$，存在一個$chi_c(G)=r$的平面圖 。Moser和Zhu的證明是蠻複雜的。然而我們的構造概念比較簡單。而且當$ngeq5$，$K_n$-minor free圖是用統一的方法構造的。
就$K_n$-minor free圖而言，雖然Hell和Zhu [9] 證明了不存在任何 的$K_n$-minor free圖$G$。對於當任意的有理數 時，是否存在一個$chi_c(G)=r$的$K_4$-minor free$G$的問題未被解決。在本論文中解決了這個問題：對任意的有理數$rin [2,8/3]$，我們將構造一的$chi_c(G)=r$的$K_4$-minor free圖$G$。
在本篇論文也研究$K_4$-minor free圖的圓色數和girth之間的關係。對任意的整數$n$，我們確定了odd girth至少是 的$K_4$-minor free圖圓色數的最大上界。而且也證明了該上界對girth=$n$的$K_4$-minor free是最好的。
Erdős的一個經典結果告訴我們，對任意的正整數 和 ，存在一個girth至少$l$且色數至少是$n$的圖$G$。用機率的方法，Zhu [39] 也證明了對任意的正整數$l$和任意的有理數$rgeq2$,存在一個girth至少$l$且$chi_(G)=r$的圖$G$。對於$rgeq3$，Nešetřil和Zhu [21] 構造出這樣的圖。當$rin(2,3)$時，對於如何構造一個girth很大且$chi_c(G)=r$的圖未有解決。在本篇論文，我們用一套統一的方法，對任意有理數$rgeq2$，構造一個girth至少$l$且$chi_c(G)=r$的圖G。
具有$chi_c(G)=chi(G)$的圖$G$在文獻中被廣泛的研究。很多種類的圖$G$已經知道滿足$chi_c(G)=chi(G)$。然而在本文之前，尚未有一個完整的方法構造一個任意大且滿足$chi_c(G)=chi(G)$且$Delta(G)$是有界的$chi$-critical圖。本篇文章給一個構造這種圖的方法。
圓流數$Phi_c(G)是$chi_c(G)對偶概念。令$G$是一個圖，把每個邊$e=xy$都換成一組相對的有向邊$a=vec{xy},a^{-1}=vec{yx}，可以得到一個對稱的有向圖。$G$的所有有向邊所組成的集合記做$A(G)$。一個chain是一的映射$f:A->R$滿足對任意的有向邊$a,f(a)=-(a^{-1})$。一個流(flow)是一個chain滿足對任意的$V(G)$的子集$X$,$sum_{ain[X,ar(X)]}f(a)=0$，而$[X,ar{X}]$是從$X$到$V-X$有向邊所組成的集合。如果一個流滿足對任意的有向邊$ain A(G)$,$1leq |f(a)|leq r-1$，則稱$f$為一個$r$流($r$-flow)。所謂$G$的圓流數$Phi_c(G)$是最小的$r$使得$G$有一個$r$流。由Tutte猜想斷言對任意的圖$G$滿足$Phi_c(G)=5$。根據平面圖的對偶圖性質，還有Moser和Zhu的結果，可以推得對任意的有理數$rin[2,4]$，存在一個$Phi_c(G)=r$的平面圖$G$。在本篇論文對任意有理數$rin(4,5)$，我們構造一個滿足$Phi_c(G)=r$的圖。

This thesis constructs special graphs with given circular
chromatic numbers or circular flow numbers.

Suppose $G=(V,E)$ is a graph and $rgeq 2$ is a real number. An
$r$-coloring of a graph $G$ is a mapping $f:V
ightarrow [0,r)$
such that for any adjacent vertices $x,y$ of $G$, $1leq
|f(x)-f(y)|leq r-1$. The circular chromatic number $chi_c(G)$
is the least $r$ for which there exists an $r$-coloring of $G$.
The circular chromatic number was introduced by Vince in 1988 in
cite{vince}, where the parameter is called the {em star
chromatic number} and denoted by $chi^*(G)$. Vince proved that
for any rational number $k/dgeq 2$ there is a graph $G$ with
$chi_c(G)=k/d$. In this thesis, we are interested in the
existence of special graphs with given circular chromatic numbers.


A graph $H$ is called a minor of a graph $G$ if $H$ can be
obtained from $G$ by deleting some vertices and edges, and
contracting some edges. A graph $G$ is called $H$-minor free if
$H$ is not a minor of G. The well-known Hadwiger's conjecture
asserts that for any positive integer $n$, any $K_n$-minor free
graph $G$ is $(n-1)$-colorable. If this conjecture is true, then
for any $K_n$-minor free graph $G$, we have $chi_c(G)leq n-1$.
On the other hand, for any graph $G$ with at least one edge we
have $chi_c(G)geq 2$. A natural question is this: Is it true
that for any rational number $2leq rleq n-1$, there exist a
$K_n$-minor free graph $G$ with $chi_c(G)=r$?

For $n=4$, the answer is ``no". It was proved by Hell and Zhu in
cite{hz98} that if $G$ is a $K_4$-minor free graph then either
$chi_c(G)=3$ or $chi_c(G)leq 8/3$. So none of the rational
numbers in the interval $(8/3,3)$ is the circular chromatic number
of a $K_4$-minor free graph. For $ngeq 5$, Zhu cite{survey}
proved that for any rational number $rin[2,n-2]$, there exists a
$K_n$-minor free graph $G$ with $chi_c(G)=r$. The question
whether there exists a $K_n$-minor free graph $G$ with
$chi_c(G)=r$ for each rational number $rin(n-2,n-1)$ remained
open. In this thesis, we answer this question in the affirmative.
For each integer $ngeq 5$, for each rational number
$rin[n-2,n-1]$, we construct a $K_n$-minor free graph $G$ with
$chi_c(G)=r$. This implies that for each $ngeq 5$, for each
rational number $rin[2,n-1]$, there exists a $K_n$-minor free
graph $G$ with $chi_c(G)=r$. In case $n=5$, the $K_5$-minor free
graphs constructed in this thesis are actually planar graphs. So
our result implies that for each rational number $rin[2,4]$,
there exists a planar graph $G$ with $chi_c(G)=r$. This result
was first proved by Moser cite{moser} and Zhu cite{3-4}. To be
precise, Moser cite{moser} proved that for each rational number
$rin[2,3]$, there exist a planar graph $G$ with $chi_c(G)=r$,
and Zhu cite{3-4} proved that for each rational number
$rin[3,4]$, there exists a planar graph $G$ with $chi_c(G)=r$.
Moser's and Zhu's proofs are quite complicated. Our construction
is conceptually simpler. Moreover, for $ngeq 5$, $K_n$-minor
free graphs, including the planar graphs are constructed with a
unified method.

For $K_4$-minor free graphs, although Hell and Zhu cite{hz98}
proved that there is no $K_4$-minor free graph $G$ with
$chi_c(G)in (8/3,3)$. The question whether there exists a
$K_4$-minor free graph $G$ with $chi_c(G)=r$ for each rational
number $rin[2,8/3]$ remained open. This thesis solves this
problem: For each rational number $rin[2,8/3]$, we shall
construct a $K_4$-minor free $G$ with $chi_c(G)=r$.

This thesis also studies the relation between the circular
chromatic number and the girth of $K_4$-minor free graphs. For
each integer $n$, the supremum of the circular chromatic number of
$K_4$-minor free graphs of odd girth (the length of shortest odd
cycle) at least $n$ is determined. It is also proved that the
same bound is sharp for $K_4$-minor free graphs of girth $n$.

By a classical result of ErdH{o}s, for any positive integers $l$
and $n$, there exists a graph $G$ of girth at least $l$ and of
chromatic number $n$. Using probabilistic method, Zhu
cite{unique} proved that for each integer $l$ and each rational
number $rgeq 2$, there is a graph $G$ of girth at least $l$ such
that $chi_c(G)=r$. Construction of such graphs for $rgeq 3$ was
given by Nev{s}etv{r}il and Zhu cite{nz}. The question of how
to construct large girth graph $G$ with $chi_c(G)=r$ for given
$rin(2,3)$ remained open. In this thesis, we present a unified
method that constructs, for any $rgeq 2$, a graph $G$ of girth
at least $l$ with circular chromatic number $chi_c(G) =r$.

Graphs $G$ with $chi_c(G)=chi(G)$ have been studied extensively
in the literature. Many families of graphs $G$ are known to
satisfy $chi_c(G)=chi(G)$. However it remained as an open
question as how to construct arbitrarily large $chi$-critical
graphs $G$ of bounded maximum degree with $chi_c(G)=chi(G)$.
This thesis presents a construction of such graphs.

The circular flow number $Phi_c(G)$ is the dual concept of
$chi_c(G)$. Let $G$ be a graph. Replace each edge $e=xy$ by a
pair of opposite arcs $a=overrightarrow{xy}$ and
$a^{-1}=overrightarrow{yx}$. We obtain a symmetric directed
graph. Denote by $A(G)$ the set of all arcs of $G$. A chain is a
mapping $f:A(G)
ightarrow I!!R$ such that for each arc $a$,
$f(a^{-1})=-f(a)$. A flow is a chain such that for each subset
$X$ of $V(G)$, $sum_{ain[X,ar{X}]}f(a)=0$, where
$[X,ar{X}]$ is the set of all arcs from $X$ to $V-X$. An
$r$-flow is a flow such that for any arc $ain A(G)$ , $1leq
|f(a)| leq r-1$. The circular flow number of $G$ is
$Phi_c(G)=mbox{ inf}{r: G mbox{ admits a } rmbox{-flow}}$.
It was conjectured by Tutte that every graph $G$ has
$Phi_c(G)leq 5$. By taking the geometrical dual of planar
graphs, Moser's and Zhu's results concerning circular chromatic
numbers of planar graphs imply that for each rational number
$rin[2,4]$, there is a graph $G$ with $Phi_c(G)=r$. The question
remained open whether for each $rin(4,5)$, there exists a graph
$G$ with $Phi_c(G)=r$. In this thesis, for each rational number
$rin [4,5]$, we construct a graph $G$ with $Phi_c(G)=r$.

Contents

1 Introduction 3
1.1 Basic concepts in graph theory………………………………………………3
1.2 Circular chromatic number………………………………………………….6
1.3 Coloring problem………………………………………………………….12
1.4 Circular flow number……………………………………………………...15
1.5 Flow problems…………………………………………………………….16
1.6 Results of this thesis………………………………………………………17
2 Labeling method 21
2.1 Label set of rooted graphs………………………………………………..21
2.2 Series and parallel joins………………………………………………….26
3 Series-parallel graphs 32
3.1 Definition of series-parallel graphs……………………………………….32
3.2 Triangle free series-parallel graphs……………………………………….35
3.3 Density of the circular chromatic number of series-parallel graphs……..37
4 Planar graphs and $K_n$-minor free graphs 44
4.1 Diamond transformation and $k^+$-series join………………………….44
4.2 Planar graphs……………………………………………………………46
4.3 $K_n$-minor free graphs4………………………………………………..51
5 Construction of large girth graphs and large critical graphs 58
5.1 Construction of large girth graphs with given circular chromatic number 58
5.2 Construction of large critical graphs of bounded maximum degree…..64
6 Series-parallel graphs of large girth 68
6.1 Main results……………………………………………………………68
6.2 Upper bounds on $chi_c(G)$ for series-parallel graphs of large odd girth.69
6.3 Lower bounds on $chi_c(G($ for series-parallel graphs of large girth..82
7 Circular flow 100
7.1 Flow conjectures and results………………………………………….101
7.2 Flow-coloring duality…………………………………………………102
7.3 Construction of graphs with given circular flow number…………….105

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