| Summary: | 碩士 === 國立嘉義大學 === 應用數學系研究所 === 100 === A graph G consists of a nonempty vertex set V(G) and an edge set E(G). All graphs considered in this thesis are finite, loopless, and without multiple edges. Let k be an integer. A (proper) k-coloring of a graph G is a mapping f : V(G) -> {1,2,...,k} such that adjacent vertices have different images. The images are called colors and all vertices of a fixed color constitute a color class. Then a k-coloring of a graph G is said to be r-equitable if the size of any two color classes differ by at most r. And, a graph G is called r-equitably k-colorable if G has an r-equitable k-coloring. Besides, the least k such that a graph G is r-equitably k-colorable is called the r-equitable chromatic number of G. Also, the least n such that a graph G is r-equitably k-colorable for all k which is bigger than n is called the r-equitable chromatic threshold of G. In fact, the notion of r-equitable colorability is a natural generalization of the well-known equitable colorability, which is the case when r = 1.
A graph G is called a bipartite graph, denoted by G(X,Y), if V(G) can be partitioned into two subsets X and Y such that every edge of G joins a vertex of X to a vertex of Y. Moreover, if every vertex of X is adjacent to every vertex of Y, then we call G(X,Y) a complete bipartite graph. Besides, let the sizes of X and Y are s and t. When s and t are equal to some positive integer n, then G(X,Y) is also called a balanced complete bipartite graph.
In this thesis, we first propose a necessary and sufficient condition for a (balanced) complete bipartite graph to be r-equitably k-colorable. Then we derive explicit formulas related to the r-equitable chromatic number and the r-equitable chromatic threshold of a (balanced) complete bipartite graph. Finally, we have some other results on the r-equitable k-coloring of a (balanced) complete bipartite graph.
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