### Title page for etd-0627103-111233

URN etd-0627103-111233 Zhi-Shi Pan panjs@math.nsysu.edu.tw This thesis had been viewed 5202 times. Download 2352 times. Applied Mathematics 2002 2 Ph.D. English Construction of Graphs with Given Circular Chrotmatic Number or Circular Flow number 2003-06-23 105 circular chromatic number circular flow number This thesis constructs special graphs with given circularchromatic 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:Vightarrow [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 incite{vince}, where the parameter is called the {em starchromatic number} and denoted by \$chi^*(G)\$. Vince proved thatfor any rational number \$k/dgeq 2\$ there is a graph \$G\$ with\$chi_c(G)=k/d\$. In this thesis, we are interested in theexistence of special graphs with given circular chromatic numbers.A graph \$H\$ is called a minor of a graph \$G\$ if \$H\$ can beobtained from \$G\$ by deleting some vertices and edges, andcontracting some edges. A graph \$G\$ is called \$H\$-minor free if\$H\$ is not a minor of G. The well-known Hadwiger's conjectureasserts that for any positive integer \$n\$, any \$K_n\$-minor freegraph \$G\$ is \$(n-1)\$-colorable. If this conjecture is true, thenfor 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 wehave \$chi_c(G)geq 2\$. A natural question is this: Is it truethat 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 incite{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 rationalnumbers in the interval \$(8/3,3)\$ is the circular chromatic numberof 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 questionwhether there exists a \$K_n\$-minor free graph \$G\$ with\$chi_c(G)=r\$ for each rational number \$rin(n-2,n-1)\$ remainedopen. 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 eachrational number \$rin[2,n-1]\$, there exists a \$K_n\$-minor freegraph \$G\$ with \$chi_c(G)=r\$. In case \$n=5\$, the \$K_5\$-minor freegraphs constructed in this thesis are actually planar graphs. Soour result implies that for each rational number \$rin[2,4]\$,there exists a planar graph \$G\$ with \$chi_c(G)=r\$. This resultwas first proved by Moser cite{moser} and Zhu cite{3-4}. To beprecise, 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 constructionis conceptually simpler. Moreover, for \$ngeq 5\$, \$K_n\$-minorfree graphs, including the planar graphs are constructed with aunified 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 rationalnumber \$rin[2,8/3]\$ remained open. This thesis solves thisproblem: For each rational number \$rin[2,8/3]\$, we shallconstruct a \$K_4\$-minor free \$G\$ with \$chi_c(G)=r\$.This thesis also studies the relation between the circularchromatic number and the girth of \$K_4\$-minor free graphs. Foreach integer \$n\$, the supremum of the circular chromatic number of\$K_4\$-minor free graphs of odd girth (the length of shortest oddcycle) at least \$n\$ is determined. It is also proved that thesame 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 ofchromatic number \$n\$. Using probabilistic method, Zhucite{unique} proved that for each integer \$l\$ and each rationalnumber \$rgeq 2\$, there is a graph \$G\$ of girth at least \$l\$ suchthat \$chi_c(G)=r\$. Construction of such graphs for \$rgeq 3\$ wasgiven by Nev{s}etv{r}il and Zhu cite{nz}. The question of howto construct large girth graph \$G\$ with \$chi_c(G)=r\$ for given\$rin(2,3)\$ remained open. In this thesis, we present a unifiedmethod that constructs, for any \$rgeq 2\$, a graph \$G\$ of girthat least \$l\$ with circular chromatic number \$chi_c(G) =r\$.Graphs \$G\$ with \$chi_c(G)=chi(G)\$ have been studied extensivelyin the literature. Many families of graphs \$G\$ are known tosatisfy \$chi_c(G)=chi(G)\$. However it remained as an openquestion as how to construct arbitrarily large \$chi\$-criticalgraphs \$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 apair of opposite arcs \$a=overrightarrow{xy}\$ and\$a^{-1}=overrightarrow{yx}\$. We obtain a symmetric directedgraph. Denote by \$A(G)\$ the set of all arcs of \$G\$. A chain is amapping \$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 planargraphs, Moser's and Zhu's results concerning circular chromaticnumbers of planar graphs imply that for each rational number\$rin[2,4]\$, there is a graph \$G\$ with \$Phi_c(G)=r\$. The questionremained 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\$. 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 indicate access worldwide 2003-06-27

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