對任意的圖G, 反魔方標號是E(G)和{1, 2, …, |E(G)|}之間的一一對應, 使得對任意兩個不同的頂點u和v, 其點和是相異的. 對任一個點 v, v的點和是指其相鄰所有的邊其邊上標號的總和.

Hartsfield和 Ringel 在1990年提出猜想: 對任意的圖G, 如果G不是兩個點的完全圖, 則G有反魔方標號. 在本篇論文中我們證明了對任意正則圖, 若其度數為奇數, 則存在反魔方標號. 另外對於樹, 我們證明在度數為二的點在滿足某些限制時, 存在反魔方標號. 最後對某些特殊類型的笛卡爾乘積圖, 我們給出了一組反魔方標號.

關鍵詞：反魔方，正則圖，樹，笛卡爾乘積圖。

An antimagic labeling of a graph G is a one-to-one correspondence between
E(G) and {1, 2, . . . , |E|} such that for any two distinct vertices u and
v, the sum of the labels assigned to edges incident to u is distinct from the
sum of the labels assigned to edges incident to v. It was conjectured by
Hartsfield and Ringel [9] in 1990 that every connected graph other than K2
has an antimagic labeling. This conjecture attracted considerable attention.
It has been verified for many special classes of graphs, such as paths, cycles,
wheels, complete graphs, dense graphs, graphs with maximum degree
≥ |V (G)| − 2, regular bipartite graphs, trees with at most one vertex of
degree 2, graphs of order pk containing a Cp-factor for some prime number p,
and the Cartesain product of paths, cycles, the Cartesian product of an arbitrary
graph with a antimagic regular graph, etc. Nevertheless, the conjecture
remained largely open.
In Chapter 2, we study antimagic labeling of regular graphs. It is proved
that if k is an odd integer, then every k-regular graph is antimagic. For
an even integer k, a sufficient condition is given for a k-regular graph to be
antimagic.
In Chapter 3, we study antimagic labeling of trees. It was proved in [12]
that if a tree T has at most one degree 2 vertex, then T is antimagic. The
proof contains a minor error. We first correct this error. Then we study trees
which contains more degree 2 vertices. Let V2 be the set of degree 2 vertices
of T. We prove that if V2 and V V2 are both independent sets, then T is
antimagic. We also prove that if the set of even degree vertices induces a
path, then T is antimagic.
In Chapter 4, we study antimagic labeling of Cartesian product of graphs.
It was proved in [25] that if G is an antimagic regular graph, then for any
graph H, G2H is antimagic. We strengthen this result by not requiring G
to be antimagic. We prove that if G is k-regular for k ≥ 2, and H 6= K1 is a connected graph, then the Cartesian product H2G is antimagic. Observe
that G2K1 is the graph G itself. So the condition that H 6= K1 is natural.
If G is 1-regular and connected, then G = K2 and K22H is called the prism
of H. We prove that if a connected graph H has at least one vertex of odd degree, then the prism of H is antimagic; if H has at least 2|V (H)| − 2 edges, then the prism of H is also antimagic. Not much is known about the Cartesian product of two non-regular graphs. We prove that if each of G1 and G2 is obtained from a regular graph by adding a universal vertex, then the Cartesian product G12G2 and the Cartesian powers of Gi are both antimagic. We also prove that if G1 and G2 are double stars, then G12G2 is antimagic.

Abstract i
Contents iii
List of Figures v
1 Introduction 1
1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Antimagic labeling of graphs . . . . . . . . . . . . . . . . . . . 4
1.3 Results of the thesis . . . . . . . . . . . . . . . . . . . . . . . 8
2 Regular graphs 9
2.1 Odd degree regular graph . . . . . . . . . . . . . . . . . . . . 9
2.2 Even degree regular graphs . . . . . . . . . . . . . . . . . . . . 19
3 Trees 28
3.1 Trees with at most one degree 2 vertex . . . . . . . . . . . . . 29
3.2 Trees with degree 2 vertices form an independent set . . . . . 33
3.3 Trees with degree 2 vertices induces a path . . . . . . . . . . . 42
4 Cartesian product of graphs 50
4.1 Cartesian product of a regular graph and an arbitrary graph . 50
4.2 The prism of graphs . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Cartesian powers . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Special Cartesian product graphs . . . . . . . . . . . . . . . . 66
5 Conclusions and further research 76

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