A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the...A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the problem of the existence of a kernel is NP-complete for a general digraph.Bang-Jensen and Gutin pose an interesting problem(Problem 12.3.5)in their book[Digraphs:Theory,Algorithms and Applications,London:Springer-Verlag,2000]:to characterize all circular digraphs with kernels.In this paper,we study the problem of the existence of the kernel for several special classes of circular digraphs.Moreover,a class of counterexamples is given for the Duchet kernel conjecture(for every connected kernel-less digraph which is not an odd directed cycle,there exists an arc which can be removed and the obtained digraph is still kernel-less).展开更多
The authors give an upper bound for the projective plane crossing number of a circular graph. Also, the authors prove the projective plane crossing numbers of circular graph C （8, 3） and C （9, 3） are 2 and 1, resp...The authors give an upper bound for the projective plane crossing number of a circular graph. Also, the authors prove the projective plane crossing numbers of circular graph C （8, 3） and C （9, 3） are 2 and 1, respectively.展开更多
Koetzig put forward a question on strongly-regular self-complementary graphs, that is, for any natural number k, whether there exists a strongLy-regular self- complementary graph whose order is 4k ＋ 1, where 4k ＋ 1 ...Koetzig put forward a question on strongly-regular self-complementary graphs, that is, for any natural number k, whether there exists a strongLy-regular self- complementary graph whose order is 4k ＋ 1, where 4k ＋ 1 = x^2 ＋ y^2, x and y are positive integers; what is the minimum number that made there exist at least two non-isomorphic strongly-regular self-complementary graphs. In this paper, we use two famous lemmas to generalize the existential conditions for strongly-regular self-complementary circular graphs with 4k ＋ 1 orders.展开更多
文摘A kernel in a directed graph D=(V,A)is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex v in V\K there is a vertex u in K,such that(v,u)is an arc of D.It is well known that the problem of the existence of a kernel is NP-complete for a general digraph.Bang-Jensen and Gutin pose an interesting problem(Problem 12.3.5)in their book[Digraphs:Theory,Algorithms and Applications,London:Springer-Verlag,2000]:to characterize all circular digraphs with kernels.In this paper,we study the problem of the existence of the kernel for several special classes of circular digraphs.Moreover,a class of counterexamples is given for the Duchet kernel conjecture(for every connected kernel-less digraph which is not an odd directed cycle,there exists an arc which can be removed and the obtained digraph is still kernel-less).
基金the National Natural Science Foundation of China under Grant No.10671073Scientific Study Foundation of the Talented People Gathered by Nantong University+2 种基金Science and Technology Commission of Shanghai Municipality under Grant No.07XD14011Shanghai Leading Academic Discipline Project under Grant No.B407Natural Science Foundation of Jiangsu's Universities under Grant No.07KJB110090
文摘The authors give an upper bound for the projective plane crossing number of a circular graph. Also, the authors prove the projective plane crossing numbers of circular graph C （8, 3） and C （9, 3） are 2 and 1, respectively.
文摘Koetzig put forward a question on strongly-regular self-complementary graphs, that is, for any natural number k, whether there exists a strongLy-regular self- complementary graph whose order is 4k ＋ 1, where 4k ＋ 1 = x^2 ＋ y^2, x and y are positive integers; what is the minimum number that made there exist at least two non-isomorphic strongly-regular self-complementary graphs. In this paper, we use two famous lemmas to generalize the existential conditions for strongly-regular self-complementary circular graphs with 4k ＋ 1 orders.
