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
The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤n) is Z(m,n). where Z(m,n) = [m/2] [(m-1)/2] [n/2] [(n-1)/2](for and real number x, [x] denotes the...The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤n) is Z(m,n). where Z(m,n) = [m/2] [(m-1)/2] [n/2] [(n-1)/2](for and real number x, [x] denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤G. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is Z(9, n) + 12[n/2].展开更多
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.
基金This work is supported by the Key Project of the Education Department of Hunan Province of China (05A037)by Scientific Research Fund of Hunan Provincial Education Department (06C515).
文摘The well known Zarankiewicz' conjecture is said that the crossing number of the complete bipartite graph Km,n (m≤n) is Z(m,n). where Z(m,n) = [m/2] [(m-1)/2] [n/2] [(n-1)/2](for and real number x, [x] denotes the maximal integer no more than x). Presently, Zarankiewicz' conjecture is proved true only for the case m≤G. In this article, the authors prove that if Zarankiewicz' conjecture holds for m≤9, then the crossing number of the complete tripartite graph K1,8,n is Z(9, n) + 12[n/2].
