网格图$P_n \times P_m$的线完成数
Line Completion Number of Grid Graph $P_n \times P_m$
摘要：
超线图的概念是在1995年由Bagga，Beineke和Varma提出的。给定一个至少有$r$条边的图，指数为$r$的超线图$L_r(G)$的顶点是$G$的$r$条边的集合，如果一个集合中的一条边与另一个集合中的一条边相邻，则两条边相邻。图$G$的行完成数$lc(G)$是使$L_r(G)$是完全图的最小正整数$r$。本文对$n$和$m$的各种情况，求出了格图$P_n \times P_m$的线完成数.
英文摘要：
The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least $r$ edges, the super line graph of index $r$, $L_r(G)$, has as its vertices the sets of $r$ edges of $G$, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number $lc(G)$ of a graph $G$ is the least positive integer $r$ for which $L_r(G)$ is a complete graph. In this paper, we find the line completion number of grid graph $P_n \times P_m$ for various cases of $n$ and $m$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
