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URN etd-0627103-111233 Author Zhi-Shi Pan Author's Email Address panjs@math.nsysu.edu.tw Statistics This thesis had been viewed 5058 times. Download 2305 times. Department Applied Mathematics Year 2002 Semester 2 Degree Ph.D. Type of Document Language English Title Construction of Graphs with Given Circular Chrotmatic Number or Circular Flow number Date of Defense 2003-06-23 Page Count 105 Keyword circular chromatic number circular flow number Abstract 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 $Phi_c(G)=r$.Advisory Committee Hung-Lin Fu - chair

D. J. Guan - co-chair

Hong-Gwa Yeh - co-chair

Li-Da Tong - co-chair

Chin-Mei Fu - co-chair

Xuding Zhu - advisor

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etd-0627103-111233.pdf Date of Submission 2003-06-27