廣義的Petersen圖的(a,d)-反魔術標號之探討

Abstract

在本篇論文中，我們討論廣義的Petersen圖的(a,d)-反魔術標號，首先我們給出一個必要條件，並用另一種方式呈現已知的定理與猜測，特別地，我們給出P(6,2)為(12,3)-反魔術圖及P(7,3)為(20,2)-反魔術圖的實際例子並且證明出P(7,2)及P(7,3)皆不為(7,4)-反魔術圖，最後我們給出一個表格，列出n=3~8時所有P(n,k)之(a,d)-反魔術圖情形，藉以猜測更大的n的反魔術標號情形。

In this thesis, we discuss (a,d)-antimagic labeling of generalized Petersen graph P(n,k). First, we give a necessary condition for the existence of P(n,k), and represent some previously known theorems in our setting. Then we show that P(6,2) has (12,3)-antimagic property and P(7,3) has (20,2)-antimagic by direct construction. Moreover, we show that neither P(7,2) nor P(7,3) is (7,4)-antimagic. Finally, we give a table showing (a,d)-antimagic property for P(n,k), when n=3~8; and conjecture that the same property holds for larger n.