依明江·沙比尔,尚辉,孟吉翔

• 1:新疆大学数学与系统科学学院

摘要(Abstract)：

图G的生成连通度为最大的正整数k使得G的任意两个顶点之间存在i (1≤i≤k)条内部不交的路,并且这些路的并生成G.文章不仅涵盖了有关图的生成连通度的最新研究进展,还包含了图的生成连通度相关的超生成连通性、生成可系性、超生成可系性等问题的最新结果.除此之外,还讨论了一些值得进一步研究的问题.

关键词(KeyWords)： 连通性;哈密顿性;生成连通性;互联网络

Abstract:

Keywords:

基金项目(Foundation): Project supported by the National Natural Science Foundation of China(11531011)

作者(Author): 依明江·沙比尔,尚辉,孟吉翔

DOI: 10.13568/j.cnki.651094.2018.04.001

参考文献(References)：

• [1]Bondy J,Murty U.Graph Theory with Applications[M].New York:Springer,2008.
• [2]Menger K.Zur allgemeinen kurventheorie[J].Fundam Math,1927,10:96-115.
• [3]Albert M,Aldred R,Holton D.On 3*-connected graphs[J].Austral J Combin,2001,24:193-208.
• [4]Hsu L,Lin C.Graph thoery and interconnection networks[M].New York:CRC Press,2008.
• [5]Dirac G.In abstrakten Graphen vorhandene vollstaandige 4-Graphen und ihre Unterteilungen[J].Mathematische Nachrichten,1960,22:61-85.
• [6]Lin C,Tan J,Hsu D,et al.On the spanning fan-connectivity of graphs[J].Discrete Appl Math,2009,157:1342-1348.
• [7]Dirac G.Some theorems on abstract graphs[J].Proc London Math Soc,1952,2:69-81.
• [8]Lin C,Huang H,Hsu L.On the spanning connectivity of graphs[J].Discrete Math,2007,307:285-289.
• [9]Ore O.Hamilton-connected graphs[J].J Math Pure Appl,1963,42:21-27.
• [10]Ore O.Note on Hamilton circutes[J].Amer Math Month,1965,67:55.
• [11]Bondy J,Chavatal V.A method in graph theory[J].Discrete Math,2008,15:111-135.
• [12]Lin C,Huang H,Tan J,et al.On spanning connected graphs[J].Discrete Math,2008,308:1330-1333.
• [13]Fan G.New sufficient cnditions for cycles in graphs[J].J Combin Theory Ser B,1984,37:221-227.
• [14]Berman A,Wojda A.The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs[J].J Combin Theory Ser B,1987,42:167-180.
• [15]Watkins M.A theorem on Tait colorings with an application to the generalized Petersen graphs[J].J Combin Theory B,1969,6:137-142.
• [16]Wang J,Hsu L.On the spanning connectivity of the generalized petersen graphs P(n,3)[J].Discrete Math,2018,341:672-690.
• [17]Kao S,Hsu H,Hsu L.Globally bi-3*-connected graphs[J].Discrete Math,2009,309:1931-1946.
• [18]Zhou J,Feng Y.Cubic vertex-transitive non-Cayley graphs of order 8p[J].Electron J Combin,2012,19:453-472.
• [19]Wang X.All doubel generalized Petersen graps are Hamiltonian[J].Discrete Math,2017,340:3016-3019.
• [20]Zhan S.Hamilton connectedness of line graphs[J].Ars Combin,1986,22:89-95.
• [21]Huang P,Hsu L.The spanning connectivity of line graphs[J].Appl Math Lett,2011,24:1614-1617.
• [22]Karaganis J.On the cube of a graph[J].Canad Math Bull,1968,11:295-296.
• [23]Sekanina M.On an order of the set of vertices of a connected graph[J].Publ Fac Sci Univ Brno,1960,412:137-142.
• [24]Sabir E,Vumar E.Spanning connectivity of the power of a graph and Hamilton-connected index of a graph[J].Graph Combin,2014,2:69-81.
• [25]Harary F,Hayes J,Wu H.A survey of the theory of hypercube graphs[J].Comput Math Appl,1988,15:277-289.
• [26]Harary F,Lewinter M.Hypercubes and other recursively defined Hamilton laceable graphs[J].Conger Numer,1987,60:81-84.
• [27]Chang C,Lin C,Huang H,et al.The subper laceability of hypercubes[J].Inf Process Lett,2004,92:15-21.
• [28]Lin C,Tan J,Hsu D,et al.On the spanning connectivity of and spanning laceability of hypercube-like networks[J].Theoret Comput Sci,2007,381:218-229.
• [29]Lin C,Ho T,Tan J,et al.Fault-tolerant Hamiltonian laceability and fault-tolerant conditional hamiltonian for bipartite hypercube-like networks[J].J Int Networks,2009,10:243-251.
• [30]Lin C,Teng Y,Tan J,et al.The spanning laceability on the faulty bipartite hypercube-like networks[J].Appl Math Comput,2013,219:8095-8103.
• [31]Choudum S,Sunitha V.Augmented cubes[J].Networks,2002,40:71-84.
• [32]Lin C,Ho T,Tan J,et al.The super spanning connectivity of augmented cubes[J].Ars combin,2012,104:161-167.
• [33]Efe K.A variation on the hypercube with lower diameter[J].IEEE Trans Comput,1991,40:1312-1316.
• [34]El-Amawy A,Latifi S.Properties and performance of folded hypercubes[J].IEEE Trans Parallel Distrib Syst,1991,2:31-42.
• [35]Hsieh S.Some edge fault-tolerant propertices of the folded hypercubes[J].Networks,2008,51:92-101.
• [36]Ma M.The spanning connectivity of folded hypercubes[J].Inf Sci,2010,180:3373-3379.
• [37]Tzeng N,Wei S.Enhanced hypercubes[J].IEEE Trans Comput,1991,40:284-294.
• [38]Chang C,Lin C,Tan J,et al.The super spanning connectivity and super spanning laceability of enhanced hypercubes[J].J Supercomput,2009,48:66-87.
• [39]Chin C,Chen H,Hsu L.Super spanning connectivity of the fully connected cubic networks[J].J Int Networks,2011,11:61-70.
• [40]Akers S,Krisnamurthy B.A group theoretic model for symmetric network[J].IEEE Trans Comput,1989,38:555-566.
• [41]Lin C,Huang H,Hsu L.The super connectivity of the pancake graphs and star graphs[J].Theoret Comput Sci,2005,339:257-271.
• [42]Lin C,Ho T,Tan J,et al.A new isomorphic definition of the crossed cube and its super spanning connectivity[J].J Int Networks,2009,10:149-166.
• [43]Meng J,Wang S.Hamiltonian property of two class of Cayley graphs[J].J Xinjiang Univ(Natural Sci Edit),1994,11(2):19-21.
• [44]Wang A,Meng J.Hamiltonian cylces in Bi-Cayley graphs of finite Abelian groups[J].J Xinjiang Univ(Natural Sci Edit),2006,23(2):156-158.
• [45]Chiang W,Chen R.The(n,k)-star graphs:a generalized star graph[J].Inf Process Lett,1995,56:259-264.
• [46]Hsu H,Lin C,Huang H.The spanning connectivity of the(n,k)-star graphs[J].Int J Found Comput,2006,17:415-434.
• [47]Day D,Tripathi A.Arrangement graphs:a class of generalized star graph[J].Inf Process Lett,1992,42:235-241.
• [48]Teng Y.The spanning connectivity of the arrangement graphs[J].J Parallel Distrib Comput,2016,98:1-7.
• [49]Fu J.Hamiltonicity of the WK-recursive networks with and without faulty nodes[J].IEEE Trans Parallel Distrib Syst,2005,16:853-865.
• [50]You L,Fan J,Han Y.Super spanning connectivity on WK-recursive networks[J].Theory Comput Sci,2018,713:42-55.
• [51]Wu J,Huang K.The balanced hypercubes:a cube-baced system for fault-tolerant applications[J].IEEE Trans Comput,1997,46:484-490.
• [52]Zhu X.A hypercube variant with small diameter[J].J Graph Theory,2017,85:651-660.
• [53]Selcuk B,Karci A.Connected cubic network graph[J].Engin Sci Techno,2017,20:934-943.
• [54]Zhang B,Yang W,Zhang S.On the spanning connectivity of tournaments[J].Discrete Appl Math,2018,239:218-222.
• [55]Thomassen C.Hamiltonian-connected tournaments[J].Combin Theory Ser B,1980,28:293-298.
• [56]Debrujin N.Some machines defined by directed graphs[J].Theory Comput Sci,1984,32:309-319.
• [57]Bermond J.The de brujin and kautz networks:a competitor for the hypercube[J].Hypercube distrib Comput,1989,15:273-279.
• [58]Li P,Wu Y.Spanning connectedness and hamiltonian thickness of graphs and interval graphs[J].Discrete Math Theor Comput Sci,2015,16:125-210.