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URN etd-0809107-161710
Author Hung-yung Chang
Author's Email Address No Public.
Statistics This thesis had been viewed 5062 times. Download 1205 times.
Department Applied Mathematics
Year 2006
Semester 2
Degree Ph.D.
Type of Document
Language English
Title Game Colourings of Graphs
Date of Defense 2007-07-20
Page Count 62
Keyword
  • f-game chromatic number
  • f-chromatic number
  • Abstract A graph function  $f$ is a mapping which assigns each graph $H$
    a positive integer $f(H)
    leq |V(H)|$ such that $f(H)=f(H')$ if $H$ and $H'$ are
    isomorphic. Given a graph function $f$ and a graph $G$, an
    $f$-colouring of $G$ is a mapping $c: V(G) o
    mathbb{N}$ such that every subgraph $H$ of $G$ receives at least
    $f(H)$ colours, i.e., $|c(H)| geq f(H)$. The $f$-chromatic
    number, $chi(f,G)$, is the minimum number of colours used in an
    $f$-colouring of $G$. The $f$-chromatic number of a graph is a
    natural generalization of the chromatic number of a graph
    introduced by Nev{s}etv{r}il and Ossena de Mendez. Intuitively,
    we would like to colour the vertices of a graph $G$ with minimum
    number of colours subject to the constraint that the number of
    colours assigned to certain subgraphs must be large enough. The
    original chromatic number of a graph and many of its
    generalizations are of this nature. For example, the chromatic
    number of a graph is the least number of colours needed to colour
    the vertices of the graph so that any subgraph isomorphic to
    $K_2$ receives $2$ colours. Acyclic chromatic number of a graph is
    the least number of colours needed to colour the vertices of the
    graph so that any subgraph isomorphic to $K_2$ receives $2$
    colours, and each cycle receives at least $3$ colours.
    This thesis studies the game version of $f$-colouring of graphs.
    Suppose $G$ is a graph and $X$ is a set of colours. Two players,
    Alice and Bob, take turns colour the vertices of $G$ with colours
    from the set $X$. A partial colouring of $G$ is legal (with respect
    to graph function $f$) if for any subgraph $H$ of $G$, the sum of
    the number of colours used in $H$ and the number of uncoloured
    vertices of $H$ is at least $f(H)$. Both Alice and Bob must colour
    legally (i.e., the partial colouring produced needs to be legal).
    The game ends if either all the vertices are coloured or there are
    uncoloured vertices but there is no legal colour for any of the
    uncoloured vertices. In the former case, Alice wins the game. In the
    latter case, Bob wins the game. The $f$-game chromatic number of
    $G$, $chi_g(f, G)$, is the least number of colours that the colour
    set $X$ needs to contain so that Alice has a winning strategy.
    Observe that if $|X| = |V(G)|$, then Alice always wins. So the
    parameter $chi_g(f,G)$ is well-defined. We define the $f$-game
    chromatic index on a graph $G$, $chi'(f,G)$, to be the $f$-game
    chromatic number on the line graph of $G$.
    A natural problem concerning the $f$-game chromatic number of graphs
    is for which graph functions $f$, the $f$-game chromatic number of
    $G$ is bounded by a constant for graphs $G$ from some natural
    classes of graphs. In case the $f$-game chromatic number of a class
    ${cal K}$ of graphs is bounded by a constant, one would like to
    find the smallest such constant. This thesis studies the $f$-game
    chromatic number or index for some special classes of graphs and for
    some special graph functions $f$. The graph functions $f$ considered
    are the following graph functions:
    1. The $d$-relaxed function, ${
    m Rel}_d$, is defined as ${
    m Rel}_d(K_{1,d+1})=2$ and ${
    m Rel}_d(H)=1$ otherwise.
    2. The acyclic function, ${
    m Acy}$, is defined as ${
    m Acy}(K_2)=2$ and ${
    m Acy}(C_n)=3$ for any $n geq 3$ and
    ${
    m Acy}(H)=1$ otherwise.
    3. The $i$-path function, ${
    m Path}_i$, is defined as ${
    m Path}_i(K_2)=2$ and
    ${
    m Path}_i(P_i)=3$ and ${
    m Path}_i(H)=1$ otherwise, where $P_i$
    is the path on $i$ vertices.
    The classes of graphs considered in the thesis are outerplanar
    graphs, forests and the line graphs of $d$-degenerate graphs. In
    Chapter 2, we discuss the acyclic game chromatic number of
    outerplanar graphs. It is proved that for any outerplanar graph $G$,
    $chi_g({
    m Acy},G) leq 7$. On the other hand, there is an
    outerplanar graph $G$ for which $chi_g({
    m Acy},G) = 6$. So the
    best upper bound for $chi_g({
    m Acy},G)$ for outerplanar graphs is
    either $6$ or $7$. Moreover, we show that for any integer $n$, there
    is a series-parallel graph $G_n$ with $chi_g({
    m Acy}, G_n)
    > n$.
    In Chapter 3, we discuss the ${
    m Path}_i$-game chromatic number
    for forests. It is proved that if $i geq 8$, then for any forest
    $F$, $chi_g({
    m Path}_i, F) leq 9$. On the other hand, if $i
    leq 6$, then for any integer $k$, there is a tree $T$ such that
    $chi_g({
    m Path}_i, T) geq k$.
    Chapter 4 studies the $d$-relaxed game chromatic indexes of
    $k$-degenerated graphs. It is proved that if $d geq 2k^2 + 5k-1$
    and $G$ is $k$-degenerated, then $chi'_{
    m g}({
    m Rel}_d,G)
    leq 2k + frac{(Delta(G)+k-1)(k+1)}{d-2k^2-4k+2}$. On the other hand,
    for any positive integer $ d leq Delta-2$, there is a tree $T$
    with maximum degree $Delta$ for which $chi'_g({
    m Rel}_d, T)
    geq frac{2Delta}{d+3}$. Moreover, we show that $chi'_g({
    m Rel}_d, G) leq
    2$ if $d geq 2k + 2lfloor frac{Delta(G)-k}{2}
    floor +1$ and
    $G$ is a $k$-degenerated graph.
    Advisory Committee
  • GERARD-JENNHWA CHANG - chair
  • TUNG-SHAN FU - co-chair
  • Wun-Seng Chou - co-chair
  • D. J. Guan - co-chair
  • Tsai-Lien Wong - co-chair
  • H. G. Yeh - co-chair
  • Li-Da Tong - co-chair
  • Xuding Zhu - advisor
  • Files
  • etd-0809107-161710.pdf
  • indicate access worldwide
    Date of Submission 2007-08-09

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