Coloring and Domination of Vertices of Graphs

• Main Authors: Jun-Lin Guo, 郭俊麟
• Other Authors: none
• Format: Others
• Language: en_US
• Published: 2013
• Online Access: http://ndltd.ncl.edu.tw/handle/14874689929139797125

博士 === 國立臺灣科技大學 === 資訊管理系 === 101 === Let $G$ be a graph with vertex set $V$. A \textbf{list assignment} of a graph $G$ is a function $\mathcal{L}$ which assigns to each $v\in V$ a list $\mathcal{L}(v)$ of colors. A \textbf{list coloring} of $G$ with list assignment $\mathcal{L}$ is a map $c:V\rightarrow\bigcup_{v\in V} \mathcal{L}(v)$ such that $c(v)\in \mathcal{L}(v)$ for all $v\in V$ and $c(u)\neq c(v)$ for any adjacent vertices $u$ and $v$ in $G$. $G$ is \textbf{$k$-choosable} if any $k$-element list assignment for it permits a list coloring of it (Erd\H{o}s--Rubin--Taylor \cite{Erdo79} and Vizing \cite{Vizi76}). Thomassen \cite{Thom95} proved that planar graphs without 3-cycles and without 4-cycles are 3-choosable. In Chapter~\ref{$3$-List-Coloring Planar Graphs of Girth 4}, we prove that planar graphs without 3-cycles and with no 4-cycle sharing an edge with another 4- or 5-cycle are 3-choosable. This strengthens Thomassen's result. A family $\mathcal{F}=\{S_1,S_2,\ldots,S_n\}$ of nonempty sets is called a \textbf{set representation} of a graph $G$ with vertex set $V=\{v_1,v_2,\ldots,v_n\}$ if $v_i$ is adjacent to $v_j$ in $G$ if and only if $S_i\cap S_j\neq \emptyset$ (Erd\H{o}s--Goodman--P\'{o}sa \cite{erdo66} and Szpilrajn--Marczewski \cite{szpi45}). If any two sets $S_i,S_j$ in $\mathcal{F}$ have the property $S_i\neq S_j$ (respectively, $S_i\nsubseteq S_j$, $|S_i|=|S_j|$, or $|S_i\cap S_j|\leqslant 1$), then $\mathcal{F}$ is called a \textbf{distinct} (respectively, \textbf{antichain}, \textbf{uniform}, or \textbf{simple}) set representation of $G$ (Bylka--Komar \cite{bylk97}, Mahadev--Wang \cite{maha99}, and Tsuchiya \cite{Tsuc90}). Let $\textbf{S}(\mathcal{F})$ stand for the union of all sets in $\mathcal{F}$. Two set representations $\mathcal{F},\mathcal{F}'$ are \textbf{isomorphic} if $\mathcal{F}$ can be obtained from $\mathcal{F}'$ by a bijective mapping from $\textbf{S}(\mathcal{F}')$ to $\textbf{S}(\mathcal{F})$. If, for a graph $G$, all set representations with minimum $|\textbf{S}(\mathcal{F})|$ of one of the various types above can be classified into $\ell$ equivalence classes of the isomorphism relation, then $\ell$ is called the \textbf{intersectable number} of that type of set representation of $G$. If $\ell=1$, $G$ is called \textbf{uniquely intersectable} with respect to the respective types of set representation (Alter--Wang \cite{alte77}). In Chapter~\ref{Intersectability of Complete Graphs and Diamond-Free Graphs}, we are concerned with the uniquely intersectability of graphs without a \textbf{diamond}, i.e. a $K_4$ with one edge deleted, as an induced subgraph with respect to simple distinct set representation. In Chapter~\ref{Intersectability of Line Graphs}, we are concerned with the intersectable number of the following types of set representation of line graphs: simple distinct, simple antichain, simple uniform, and simple distinct uniform. For two natural numbers $n$ and $k$ with $n\geqslant 3$ and $1\leqslant k \leqslant \left\lfloor \frac{n-1}{2}\right\rfloor$, the \textbf{generalized Petersen graph} $P(n,k)$ is a graph on $2n$ vertices with $V(P(n,k))=\{u_i,v_i|1\leqslant i\leqslant n\}$ and $E(P(n,k))=\{u_iu_{i+1},u_iv_i,v_iv_{i+k}|1\leqslant i\leqslant n\}$ with subscripts modulo $n$ (Biggs \cite{Bigg93}, Coxeter \cite{Coxe50}, and Watkins \cite{Watk69}). A set $S\subseteq V$ is a \textbf{dominating set} of a garph $G$ if all vertices in $V\setminus S$ are adjacent to some member of $S$ (Berge \cite{Berg62}). The \textbf{domination number} of $G$, denoted by $\gamma(G)$, is the cardinality of a minimum dominating set. Ebrahimi--Jahanbakht--Mahmoodian \cite{Ebra09}, Fu--Yang--Jiang \cite{Fu09}, and Yan--Kang--Xu \cite{Yan09} independently proved that $\gamma(P(n,2))=\left\lceil \frac{3n}{5} \right\rceil$ for $n\geqslant 3$. In Chapter~\ref{Chapter 4}, we investigate the domination number of generalized Petersen graphs $P(n, 2)$ when there is a faulty vertex. Denote by $\gamma(P(n, 2))$ the domination number of $P(n, 2)$ and $\gamma(P_f(n, 2))$ the domination number of $P(n, 2)$ with a faulty vertex $u_f$. We show that $\gamma(P_f(n, 2))=\gamma(P(n, 2))-1$ when $n = 5k + 1$ or $5k + 2$ and $\gamma(P_f(n, 2))=\gamma(P(n, 2))$ for the other cases.