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URN etd-0809107-161710 Author Hung-yung Chang Author's Email Address No Public. Statistics This thesis had been viewed 5202 times. Download 1229 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

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etd-0809107-161710.pdf Date of Submission 2007-08-09