on Graph Labeling Problems of Antimagic Type

• Main Authors: Cheng-Chih Hsiao, 蕭丞志
• Other Authors: Tao-Ming Wang
• Format: Others
• Language: en_US
• Published: 2006
• Online Access: http://ndltd.ncl.edu.tw/handle/93438938170089870883

碩士 === 東海大學 === 數學系 === 94 === Assume G is a simple graph with p vertices and q edges. An edge labeling of a graph G is an assignment of integers to edges, which satisfies certain prescribed conditions. If the vertex sums are pairwise distinct in certain sense where the vertex sum is the sum of the labels of all edges incident with the vertex, we call them antimagic-type labeling. In this thesis, we study two kinds of antimagic-type labeling problems, antimagic labeling and k-edge-graceful labeling. An antimagic labeling is a bijection from the set of all edges to the set of 1,2,... ,q, such that the vertex sums are pairwise distinct. And a k-edge-graceful labeling of a graph G is a bijection from the set of all edges to the set of k,k+1,... ,k+q-1, such that the vertex sums modulo p are pairwise distinct. Our main results in this thesis are composed of two parts. The first part is about results for antimagic labeling of graphs. We study the Cartesian product of graphs, lexicographic product of graphs, and miscellaneous graphs using the methods of induction, graph decompositions, and results in magic square etc. The second part is results for k-edge-graceful labeling. We study recently the k-edge-graceful labeling of square of paths with Sin-Min Lee. While working on this topic, we found that there are close connections among graph labeling, graph decompositions, and integer sequences. This provides with more techniques to study graph labeling problems. This thesis is organized as follows : The first chapter gives the introduction to graph labeling. Main results are in the chapter two, three, and four. The second chapter deals with the antimagic labeling of Cartesian product of graphs, such as P_m ╳ P_n, C_m ╳P_n, and C_m ╳ C_n etc. The third chapter deals with the antimagic labeling of lexicographic product of graphs, join of graphs, corona of graphs. The fourth chapter deals with the edge-graceful spectrum of square of paths, and the last chapter is the conclusion.