Study on Graph Labeling and De ciencyProblems of Edge-Antimagic Type

• Main Authors: Chih-Hsiuan Liu, 劉志璿
• Other Authors: 賈明益
• Format: Others
• Language: en_US
• Published: 2018
• Online Access: http://ndltd.ncl.edu.tw/handle/xt8hwj

博士 === 國立中興大學 === 應用數學系所 === 106 === A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. A finite simple graph G = (V,E) is called (a,d)edge-antimagic if there exists an injective vertex labeling function f : V → {1,2,··· ,|V|}such that the edge labels{w(e) : e ∈ E} = {a,a+d,a+2d,······ ,a+ (|E|−1)d}, where d is a positive integer and the induced edge label is defined by w(e) = f(u)+f(v) for each e = uv ∈ E. One may consider the notion deficiency as a generalization of that of the graph labeling, which is a parameter measuring how far a graph is away from the existence of the specific labeling. Note that if the graph does not admit the labeling, then the deficiency is zero. It turns out the deficiency problems of the well known graceful labeling are closely related to the Golomb rulers with engineering applications. Therefore the study of the graph deficiency problems of antimagic types hopefully contributes to more applications. The associated (a,d) edge-antimagic deficiency µd(G) of a graph G = (V,E), which is defined as the minimum integer k such that G is (a,d)-edge-antimagic by modifying the range of the injective vertex labeling function from{1,2,··· ,|V|}to{1,2,··· ,|V|+k}. More generally a graph is called edge-antimagic if there exists an injective vertex labeling function f : V →{1,2,··· ,|V|} such that the induced edge labels w(e) 6= w(e0) for any two edges e,e0 ∈ E. The associated edge-antimagic deficiency µ(G) of a graph G is similarly defined as the minimum integer k such that G is edge-antimagic by modifying the range of the injective vertex labeling function from {1,2,··· ,|V|} to {1,2,··· ,|V|+ k}. In this dissertation, among others we completely determine the (a,d)-edge-antimagic deficiency µd, d ≥ 1, and edge-antimagic deficiency µ of complete bipartite graphs Km,n. Also we extend the study of deficiency problems to complete graphs and complete multi-partite graphs. More open problems and future studies are posted.