Title page for etd-0627103-111233


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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 5098 times. Download 2312 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
  • Files
  • etd-0627103-111233.pdf
  • indicate access worldwide
    Date of Submission 2003-06-27

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