## Domination numbers, three-arc graph construction and symmetric graphs

### Abstract

© 2012 Dr. Guangjun XuThis thesis deals with domination theory for graphs, $3$-arc graph construction and imprimitive symmetric graphs. In the first part of the thesis, we study power domination, clique transversal number, $k$-tuple domination and rainbow domination in graphs. For power domination problem, we propose a linear-time algorithm for block graphs, and as a byproduct of our algorithm we present a sharp upper bound for such graphs and characterize all graphs attaining this bound. For the class of generalized Petersen graphs, we present both upper bounds for general case and exact results for a subfamily of graphs in this class. We introduce the concept of minus clique transversal problem, and study its fundamental properties. For $k$-tuple domination, we solve a conjecture regarding an upper bound which generalizes the known upper bounds on usual domination, $2$-tuple domination and $3$-tuple domination numbers. For rainbow domination, we answer an open question. In the second part of this thesis, we explore certain combinatorial properties of the $3$-arc graph construction, including domination, coloring and Hamiltonicity. We present some sharp bounds on domination and chromatic numbers for $3$-arc graphs, introduce a new concept of arc-coloring to study the coloring problem. We then study the Hamiltonicity of $3$-arc graphs. We prove that any connected $3$-arc graph is Hamiltonian, and all iterative $3$-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative $3$-arc graphs. In addition, we show that every connected $3$-arc graph admits a second Hamilton cycle, and prove that the $3$-arc graphs of all cubic bipartite graphs have Hamiltonian decompositions. In the last part, we investigate imprimitive symmetric graphs by using a geometric approach introduced by Gardiner and Praeger in 1995. (A $G$-symmetric graph $\Gamma$ is said to be imprimitive if the group $G$ acts imprimitively on the vertices of $\Gamma$, which means that the vertex set of$\Gamma$ admits a nontrivial $G$-invariant partition ${\cal B}$.) According to this approach, three configurations are associated with $(\Gamma, {\cal B})$, namely, the quotient graph $\Gamma_{{\cal B}}$ of $\Gamma$ with respect to ${\cal B}$, the bipartite subgraph $\Gamma[B,C]$ of $\Gamma$ induced by two adjacent blocks $B,C$ of ${\cal B}$ and a certain 1-design ${\cal D}(B)$ induced on $B$. We answer an open question proposed in the case where the block size $k$ of ${\cal D}(B)$ is two less than the block size $v$ of ${\cal B}$. We also study the case where $v-k$ is an odd prime and give necessary conditions for $\Gamma_{{\cal B}}$ to be $(G, 2)$-arc transitive (regardless of whether $\Gamma$ is $(G, 2)$-arc transitive). We prove further that if $p=3$ or $5$ then these necessary conditions are essential sufficient for $\Gamma_{{\cal B}}$ to be $(G, 2)$-arc transitive

Topics: graph, domination number, 3-arc graph construction, imprimitive symmetric graph
Year: 2012
OAI identifier: oai:jupiter.its.unimelb.edu.au:11343/37316