2类与4-圈有关图的优美性

Gracefulness of Two Types of Graphs Related to Cycles of Length Four

把顺序有一个公共点的n个4圈的并图记作Fn,4;图Fn,4每个4圈的顶点ui1与ui2之间连接m条长为2的路ui1vijui2(i,j=1,2,…,n)得到的图记为m-Fn,4;将孤立顶点w与m-Fn,4的每个顶点连接一条边得到的图记为G,将图G的顶点w加n(m+1)条悬挂边所得到的图记为m-Fn,4+En(m+1).用构造的方法给出图m-Fn,4和m-Fn,4+En(m+1)的优美标号,并证明了m-Fn,4和m-Fn,4+En(m+1)都是优美图.

The graph marked Fn,4that is formed by n pieces of 4-cycles with one common vertice in order.Let ui1 and ui2be two vertices in graph Fn,4,which connect ui1 and ui2 with mpaths of length 2for ui1vijui2（i,j=1,2,…,n）.The resulting graphs are denoted as m-Fn,4.Let Grepresent the graph obtained by connecting every vertex of graph m-Fn,4isolated vertex wto an edge of it.Let m-Fn,4＋En（m＋1）be the graph obtained fromGby attaching n（m＋1）pendent edges to the vertex wof graph G.The constructor method gives graceful labeling of m-Fn,4 and m-Fn,4 ＋En（m＋1）,thus proving graceful graphs m-Fn,4and m-Fn,4＋En（m＋1）.Thus,a series of graceful graphs can be constructed from one graceful graph.