λ重完全二部3-一致超图的紧六圈分解  

Tight 6-Cycle Decompositions of λ-Fold Complete Bipartite 3-Uniform Hypergraphs

作　　者：赵红涛[1] 王静明 ZHAO Hongtao;WANG Jingming(School of Mathematics and Physics,North China Electric Power University,Beijing 102206,China)

机构地区：[1]华北电力大学数理学院,北京102206

出　　处：《昆明理工大学学报（自然科学版）》2023年第5期232-238,共7页Journal of Kunming University of Science and Technology(Natural Science)

基　　金：国家自然科学基金项目(61972148,11771140)。

摘　　要：研究了一类特殊类型的超图分解问题,即λ重完全二部3-一致超图λK_(n,n)^((3))的紧6圈分解,记作S_(λ)(3,C_(6)^((3)),n,n).首先,通过计算得到其分解存在的必要条件为λn~2(n-1)≡0(mod 6),且n≥3.故对于λ=1的情况,只需考虑n≡0,3,4,7 (mod 6);对于λ=3的情况,只需考虑n≡5,8(mod 6).然后,建立递归构造,找到递归构造中所需要的小阶数设计,针对阶数比较大的超图,可以由一些阶数较小的设计递归生成.最后得到S_(λ)(3,C_(6)^((3)),n,n)存在的必要条件也是充分的,即存在S_(λ)(3,C_(6)^((3)),n,n)当且仅当λn~2(n-1)≡0 (mod 6),且n≥3.We investigate a specific type of hypergraph decomposition problem,namely,the decomposition ofλ-fold complete bipartite 3-uniform hypergraph λK_(n,n)~((3)) into tight 6-cycles.The decomposition is denoted by S_λ(3,C_6~((3)),n,n).Firstly,we obtain that the necessary condition for the existence of the decomposition is λn~2(n-1)≡0(mod 6) and n≥3.Therefore,for the caseλ=1,we only need to consider n≡0,3,4,7 (mod 6);for the case λ=3,we only need to consider n≡5,8 (mod 6).Then,we give some recursive constructions and find the designs with small order needed in recursive constructions.For the case where hypergraphs of large order,they can be recursively generated by some designs with small orders.Finally,the necessary conditions for the existence of S_λ(3,C_6~((3)),n,n) are also sufficient.Accordingly,an S_λ(3,C~((3))_6,n,n) exists if and only if λn~2(n-1)≡0 (mod 6) and n≥3.

关 键 词：超图分解 完全二部3-一致超图 紧6圈分解 递归构造 基区组 

分 类 号：O157.5[理学—数学]