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Science China Mathematics 2003年第1期46卷 91-93页

作者：王恺顺 Department of Mathematics Beijing Normal University

In this paper, it is proved that the girth of a 4-homogeneous bipartite graph with valency greaterthan 2 is at most 12.

关键词： 4-homogeneous graph, distance-regular graph, girth.

Totally Coded Method for Signal Flow graph Algorithm

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Journal of Donghua University(English Edition) 2002年第2期19卷 63-68页

作者：徐静波周美华 College of Electronic & Electrical Engineering Shanghai University of Engineering Science Shanghai 200336 Environmental Science and Engineering College Dong Hua University Shanghai 200051

After a code-table has been established by means of node association information from signal flow graph, the totally coded method (TCM) is applied merely in the domain of code operation beyond any figure-earching algorithm. The code-series (CS) have the holo-information nature, so that both the content and the sign of each gain-term can be determined via the coded method. The principle of this method is simple and it is suited for computer programming. The capability of the computer-aided analysis for switched current network (SIN) can be enhanced.

关键词： Signal flow graph, algorithm, coded method, SIN.

Optimization problems of the third edge-connectivity of graphs

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Science China Mathematics 2006年第6期49卷 791-799页

作者： WANG Yingqian Department of Mathematics Zhejiang Normal University Jinhua 321004 China

The third edge-connectivity λ3(G) of a graph G is defined as the minimum cardinality over all sets of edges, if any, whose deletion disconnects G and each component of the resulting graph has at least 3 vertices. An upper bound has been established for λ3(G) whenever λ3(G) is well-defined. This paper first introduces two combinatorial optimization concepts, that is, maximality and superiority, of λ3(G), and then proves the Ore type sufficient conditions for G to be maximally and super third edge-connected. These concepts and results are useful in network reliability analysis.

关键词： graph, third edge-connectivity, maximality, superiority.

On semisymmetric cubic graphs of order 6p^(2)

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Science China Mathematics 2004年第1期47卷 1-17页

作者： LU Zaiping, WANG Changqun & XU MingyaoLMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China Department of Mathematics, Zhengzhou University, Zhengzhou 450052, ChinaPresent address: Department of Mathematics, Qufu Normal University, Qufu 273165, China. LMAM School of Mathematical Sciences Peking University Beijing China Department of Mathematics Zhengzhou University Zhengzhou China

A graph Γ is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Γ) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call Γ a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.

关键词： (G-)semisymmetric graph, symmetric graph, bi-Cayley graph, regular covering, group action.

Pebbling numbers of some graphs

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Science China Mathematics 2002年第4期45卷 470-478页

作者：冯荣权 Ju Young Kim School of Mathematical Sciences Peking University Beijing 100871 China

Chung defined a pebbling move on a graph G as the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph G, f(G), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). In the present paper the pebbling numbers of the product of two fan graphs and the product of two wheel graphs are computed. As a corollary, Graham's conjecture holds when G and H are fan graphs or wheel graphs.

关键词： pebbling, Graham's conjecture, Cartesian product, fan graph, wheel graph.

D(β)-vertex-distinguishing total coloring of graphs

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Science China Mathematics 2006年第10期49卷 1430-1440页

作者： ZHANG Zhongfu,LI Jingwen,CHEN Xiang’en,YAO Bing, WANG Wenjie & QIU Pengxiang Institute of Applied Mathematic, Lanzhou Jiaotong University, Lanzhou 730070, China College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China College of Information and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China Institute of Applied Mathematic Lanzhou Jiaotong University Lanzhou China College of Mathematics and Information Science Northwest Normal University Lanzhou China College of Information and Electrical Engineering Lanzhou Jiaotong University Lanzhou China

A new concept of the D(β)-vertex-distinguishing total coloring of graphs, i.e., the proper total coloring such that any two vertices whose distance is not larger than β have different color sets, where the color set of a vertex is the set composed of all colors of the vertex and the edges incident to it, is proposed in this paper. The D(2)-vertex-distinguishing total colorings of some special graphs are discussed, meanwhile, a conjecture and an open problem are presented.

关键词： graph, total coloring, D(β)-vertex-distinguishing total coloring, D(β)-vertexdistinguishing total chromatic number.

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

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Science China Mathematics 2004年第4期47卷 593-604页

作者： XU Mingyao XU Shangjin LMAM&School of Mathematical Sciences Peking UniversityBeijing 100871China

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal;that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.

关键词： Cayley graph, m-CI-group, normal Cayley graph, arc-transitive graph, GRR of a group, halftransitive graph.

On adjacent-vertex-distinguishing total coloring of graphs

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Science China Mathematics 2005年第3期48卷 289-299页

作者： ZHANG Zhongfu, CHEN Xiang’en, LI Jingwen, YAO Bing, LU Xinzhong & WANG Jianfang College of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, China Department of Computer, Lanzhou Normal College, Lanzhou 730070, China Institute of Applied Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China College of Information and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100080, China College of Mathematics and Information Science Northwest Normal University Lanzhou China Department of Computer Lanzhou Normal College Lanzhou China Institute of Applied Mathematics Lanzhou Jiaotong University Lanzhou China College of Information and Electrical Engineering Lanzhou Jiaotong University Lanzhou China Institute of Applied Mathematics Chinese Academy of Sciences Beijing China

In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number... 详细信息

关键词： graph, proper total coloring, adjacent-vertex-distinguishing total coloring, adjacent-vertex-distinguishing total chromatic number.

The proof of Ushio's conjecture concerning path factorization of complete bipartite graphs

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Science China Mathematics 2006年第3期49卷 289-299页

作者： DU Beiliang & WANG Jian Department of Mathematics, Suzhou University, Suzhou 215006, China Nantong Vocational College, Nantong 226007, China Department of Mathematics Suzhou University Suzhou China Nantong Vocational College Nantong China

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A Pv-factorization of Km,n is a set of edge-disjoint Pv-factors of Km,n which partition the set of edges of Km,n. When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of Pv-factorization of Km,n. When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio's conjecture is true when v = 4k - 1. In this paper we shall show that Ushio Conjecture is true when v = 4k + 1, and then Ushio's conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P4k+1-factorization of Km,n is (i) 2km ≤ (2k + 1)n,(ii) 2kn ≤ (2k + 1)m, (iii) m + n ≡ 0 (mod 4k + 1), (iv) (4k + 1)mn/[4k(m + n)] is an integer.

关键词： complete bipartite graph, factorization, Ushio Conjecture.

Classifying cubic symmetric graphs of order 10p or 10p^(2)

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Science China Mathematics 2006年第3期49卷 300-319页

作者： KWAK Jin Ho Department of Mathematics

A graph is called s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular cyclic or elementary abelian coverings of the Petersen graph for each s ≥ 1 are classified when the fibre-preserving automorphism groups act *** an application of these results, all s-regular cubic graphs of order 10p or 10p2 are also classified for each s ≥ 1 and each prime p, of which the proof depends on the classification of finite simple groups.

关键词： s-regular graph, regular covering, Petersen graph.