| Summary: | 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.
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