The Typenumber of the Lattice Graph

• Main Authors: Yen-Chi Chen, 陳彥吉
• Other Authors: Hung-Lin Fu
• Format: Others
• Language: en_US
• Published: 1994
• Online Access: http://ndltd.ncl.edu.tw/handle/02346763021534471123

碩士 === 國立交通大學 === 應用數學研究所 === 82 === The $type$ of a vertex $v$ in a $p$-page book-embedding is the $p\times 2$ matrix of nonnegative integers \[ \begin{array}{c} \tau (v)\\ \end{array} = \left[ \begin{array}{cc} l_{v,1}& r_{v,1}\\ \vdots & \vdots \\ l_{v,p}&r_{v,p}\\ \end{array} \ right], \] where $l_{v,i}$(respectively, $r_{v,i}$) is the number of edges incident to $v$ that connect on page $i$ to vertices lying to the left (respectively, to the right) of $v$. The $ typenumber $ of a graph $G,$ $T(G)$, is the minimum number of different types among all the book-embeddings of G. In this thesis, we shall consider the general properties of $typenumber$ and the $typenumber$ of some specific graphs. Mainly, we find an upper bound for lattice graph, $L_{m,n}$, and give exact solutions when $m\le 3$. In case that $m=2$ and $n\geq 3$, our result disprove the conjecture by J. Buss et.al. which says for $n\ge 4$, $T(L_{2,n})$ is not less than 5.