### Title page for etd-0809107-161710

URN etd-0809107-161710 Hung-yung Chang No Public. This thesis had been viewed 5218 times. Download 1246 times. Applied Mathematics 2006 2 Ph.D. English Game Colourings of Graphs 2007-07-20 62 f-game chromatic number f-chromatic number 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'\$ areisomorphic. Given a graph function \$f\$ and a graph \$G\$, an\$f\$-colouring of \$G\$ is a mapping \$c: V(G) omathbb{N}\$ such that every subgraph \$H\$ of \$G\$ receives at least\$f(H)\$ colours, i.e., \$|c(H)| geq f(H)\$. The \$f\$-chromaticnumber, \$chi(f,G)\$, is the minimum number of colours used in an\$f\$-colouring of \$G\$. The \$f\$-chromatic number of a graph is anatural generalization of the chromatic number of a graphintroduced by Nev{s}etv{r}il and Ossena de Mendez. Intuitively,we would like to colour the vertices of a graph \$G\$ with minimumnumber of colours subject to the constraint that the number ofcolours assigned to certain subgraphs must be large enough. Theoriginal chromatic number of a graph and many of itsgeneralizations are of this nature. For example, the chromaticnumber of a graph is the least number of colours needed to colourthe vertices of the graph so that any subgraph isomorphic to\$K_2\$ receives \$2\$ colours. Acyclic chromatic number of a graph isthe least number of colours needed to colour the vertices of thegraph 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 coloursfrom the set \$X\$. A partial colouring of \$G\$ is legal (with respectto graph function \$f\$) if for any subgraph \$H\$ of \$G\$, the sum ofthe number of colours used in \$H\$ and the number of uncolouredvertices of \$H\$ is at least \$f(H)\$. Both Alice and Bob must colourlegally (i.e., the partial colouring produced needs to be legal).The game ends if either all the vertices are coloured or there areuncoloured vertices but there is no legal colour for any of theuncoloured vertices. In the former case, Alice wins the game. In thelatter case, Bob wins the game. The \$f\$-game chromatic number of\$G\$, \$chi_g(f, G)\$, is the least number of colours that the colourset \$X\$ needs to contain so that Alice has a winning strategy.Observe that if \$|X| = |V(G)|\$, then Alice always wins. So theparameter \$chi_g(f,G)\$ is well-defined. We define the \$f\$-gamechromatic index on a graph \$G\$, \$chi'(f,G)\$, to be the \$f\$-gamechromatic number on the line graph of \$G\$.A natural problem concerning the \$f\$-game chromatic number of graphsis for which graph functions \$f\$, the \$f\$-game chromatic number of\$G\$ is bounded by a constant for graphs \$G\$ from some naturalclasses of graphs. In case the \$f\$-game chromatic number of a class\${cal K}\$ of graphs is bounded by a constant, one would like tofind the smallest such constant. This thesis studies the \$f\$-gamechromatic number or index for some special classes of graphs and forsome special graph functions \$f\$. The graph functions \$f\$ consideredare 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 outerplanargraphs, forests and the line graphs of \$d\$-degenerate graphs. InChapter 2, we discuss the acyclic game chromatic number ofouterplanar graphs. It is proved that for any outerplanar graph \$G\$,\$chi_g({m Acy},G) leq 7\$. On the other hand, there is anouterplanar graph \$G\$ for which \$chi_g({m Acy},G) = 6\$. So thebest upper bound for \$chi_g({m Acy},G)\$ for outerplanar graphs iseither \$6\$ or \$7\$. Moreover, we show that for any integer \$n\$, thereis 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 numberfor 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 \$ileq 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) leq2\$ if \$d geq 2k + 2lfloor frac{Delta(G)-k}{2} floor +1\$ and\$G\$ is a \$k\$-degenerated graph. 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 indicate access worldwide 2007-08-09

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