With its comprehensive application in network information engineering （e. g. dynamic spectrum allocation under different distance constraints ） and in network combination optimization （e. g. safe storage of deleter...With its comprehensive application in network information engineering （e. g. dynamic spectrum allocation under different distance constraints ） and in network combination optimization （e. g. safe storage of deleterious materials）, the graphs＇ cloring theory and chromatic uniqueness theory have been the forward position of graph theory research. The later concerns the equivalent classification of graphs with their color polynomials and the determination of uniqueness of some equivalent classification under isomorphism. In this paper, by introducing the concept of chromatic normality and comparing the number of partitions of two chromatically equivalent graphs, a general numerical condition guarenteeing that bipartite graphs K （ m, n） - A （A belong to E（K （m, n） ） and ｜ A ｜≥ 2） is chromatically unique was obtained and a lot of chromatic uniqueness graphs of bipartite graphs K（m, n） - A were determined. The results obtained in this paper were general. And the results cover and extend the majority of the relevant results obtained within the world.展开更多
Let P（G,λ） be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P（H,λ） = P（G,λ） implies that H is isomorphic to G. Many sufficient conditions guarantee...Let P（G,λ） be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P（H,λ） = P（G,λ） implies that H is isomorphic to G. Many sufficient conditions guaranteeing that some certain complete tripartite graphs are chromatically unique were obtained by many scholars. Especially, in 2003, Zou Hui-wen showed that if n 〉 1/3m2 ＋ 3/1k2 ＋ 3/1mk＋ 1/3m-1/3k＋ 3/2√m2 ＋ k2 ＋ mk, where n,k and m are non-negative integers, then the complete tripartite graph K（n - m,n,n ＋ k） is chromatically unique （or simply χ–unique）. In this paper, we prove that for any non-negative integers n,m and k, where m ≥ 2 and k ≥ 0, if n ≥ 3/1m2 ＋ 3/1k2 ＋ 3/1mk ＋ 3/1m - 3/1k ＋ 43, then the complete tripartite graph K（n - m,n,n ＋ k） is χ–unique, which is an improvement on Zou Hui-wen’s result in the case m ≥ 2 and k ≥ 0. Furthermore, we present a related conjecture.展开更多
基金Natural Science Foundation of Fujian, China (No.S0650011)
文摘With its comprehensive application in network information engineering （e. g. dynamic spectrum allocation under different distance constraints ） and in network combination optimization （e. g. safe storage of deleterious materials）, the graphs＇ cloring theory and chromatic uniqueness theory have been the forward position of graph theory research. The later concerns the equivalent classification of graphs with their color polynomials and the determination of uniqueness of some equivalent classification under isomorphism. In this paper, by introducing the concept of chromatic normality and comparing the number of partitions of two chromatically equivalent graphs, a general numerical condition guarenteeing that bipartite graphs K （ m, n） - A （A belong to E（K （m, n） ） and ｜ A ｜≥ 2） is chromatically unique was obtained and a lot of chromatic uniqueness graphs of bipartite graphs K（m, n） - A were determined. The results obtained in this paper were general. And the results cover and extend the majority of the relevant results obtained within the world.
基金Supported by the National Natural Science Foundation of China (Grant No.10771091)the Science and Research Project of the Education Department of Gansu Province (Grant No.0501-02)
文摘Let P（G,λ） be the chromatic polynomial of a simple graph G. A graph G is chromatically unique if for any simple graph H, P（H,λ） = P（G,λ） implies that H is isomorphic to G. Many sufficient conditions guaranteeing that some certain complete tripartite graphs are chromatically unique were obtained by many scholars. Especially, in 2003, Zou Hui-wen showed that if n 〉 1/3m2 ＋ 3/1k2 ＋ 3/1mk＋ 1/3m-1/3k＋ 3/2√m2 ＋ k2 ＋ mk, where n,k and m are non-negative integers, then the complete tripartite graph K（n - m,n,n ＋ k） is chromatically unique （or simply χ–unique）. In this paper, we prove that for any non-negative integers n,m and k, where m ≥ 2 and k ≥ 0, if n ≥ 3/1m2 ＋ 3/1k2 ＋ 3/1mk ＋ 3/1m - 3/1k ＋ 43, then the complete tripartite graph K（n - m,n,n ＋ k） is χ–unique, which is an improvement on Zou Hui-wen’s result in the case m ≥ 2 and k ≥ 0. Furthermore, we present a related conjecture.
