局部扭立方的1好邻连通度和诊断度（英文）

Diagnosability of a multiprocessor system is one important study topic. In 2012, Peng et al. proposed the g-good-neighbor diagnosability that restrains every fault-free node to contain at least g fault-free neighbors. The locally twisted cube LTQ_n has many good properties. In this paper, we show that the 1-good-neighbor connectivity κ~1(LTQ_n) = 2n-2 and the 1-good-neighbor diagnosability of LTQ_n is 2n-1 under the PMC model for n ≥ 4 and the MM~*model for n ≥ 5.     

Competition Numbers and Phytogeny Numbers of Connected Graphs and Hypergraphs

Let D be a digraph.The competition graph of D is the graph having the same vertex set with D and having an edge joining two different vertices if and only if they have at least one common out-neighbor in D.The phylogeny graph of D is the competition graph of the digraph constructed from D by adding loops at all vertices.The competition/phylogeny number of a graph is the least number of vertices to be added to make the graph a competition/phylogeny graph of an acyclic digraph.In this paper,we show that for any integer k there is a connected graph such that its phylogeny number minus its competition number is greater than k.We get similar results for hypergraphs.     

Distance Between α-Orientations of Plane Graphs by Facial Cycle Reversals

Cycle reversal had been shown as a powerful method to deal with the relation among orientations of a graph since it preserves the out-degree of each vertex and the connectivity of the orientations. A facial cycle reversal on an orientation of a plane graph is an operation that reverses all the directions of the edges of a directed facial cycle. An orientation of a graph is called an α-orientation if each vertex admits a prescribed out-degree. In this paper, we give an explicit formula for the minimum number of the facial cycle reversals needed to transform one α-orientation into another for plane graphs.     

三正则图的至多单折纵横嵌入

This paper provides the complete proof of the fact that any planar cubic graph isat most slngle-bend embeddable except for the tetrabedrou. An O(n) amortized time algorithm for drawing an at most single-bend embedding of a cubic graph‘.is also presented, where n is the number of vertices of the graph. Furthermore, it is proved that the minimum of the total number of bends in an at most slngle-bend embedding of a cubic graph of order n is less than or equal to 0.5n 1. This result is the best possible.     

本刊英文版Vol.21(2005),No.2论文摘要

<正>Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration Lin Fan MAO Yan Pei 1IU Feng TIAN A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph T to be an automorphism of a map with the underlying graph Γis obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces     

SUPER EDGE-CONNECTIVITY OF DE BRUIJN AND KAUTZ UNDIRECTED GRAPHS

The super edge-connectivity of a graph is an important parameter to measure fault-tolerance of interconnection networks. This note shows that the Kautz undirected graph is super edge-connected,and provides a short proof of Lue and Zhang‘s result on super edge-connectivity of the de Bruijn undirected graph.     

A Rao-type characterization for a sequence to have a realization containing an arbitrary subgraph H

Let G be an arbitrary spanning subgraph of the complete graph Kr+1 on r+1 vertices and Kr+1-E(G) be the graph obtained from Kr+1 by deleting all edges of G.A non-increasing sequence π=(d1,d2,...,dn) of nonnegative integers is said to be potentially Kr+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains Kr+1-E(G) as a subgraph.In this paper,a characterization of π that is potentially Kr+1-E(G)-graphic is given,which is analogous to the Erdo s–Gallai characterization of graphic sequences using a system of inequalities.This is a solution to an open problem due to Lai and Hu.As a corollary,a characterization of π that is potentially Ks,tgraphic can also be obtained,where Ks,t is the complete bipartite graph with partite sets of size s and t.This is a solution to an open problem due to Li and Yin.     

A NEW APPROACH TO THE LINEARITY OF TESTING PLANARITY OF GRAPHS

In testing planarity of graphs,there are many criteria.The earliest one as known is the Kuratowski's theorem,then Whitney's,Maclane's, and so forth.Since the early sixties,people have begun researches on algorithms.Up to 1974,Hopcroft and Tarjan found an algorithm with a computing time being a linear function of the order of a graph.This is the linearity concerned here.This paper presents a new approach to the linearity by means of transforming the problem of testing planarity of a graph G into that of finding a spanning tree on another graph H,called an auxiliary graph of G,with the order of H being a linear function of that of G.And moreover,we can also make the size of H be a linear function of that of G.The whole procedure is based on the building up of a theory of linear equations on GF(2)related to G.     

Graphs isomorphic to their maximum matching graphs

The maximum matching graph M（G） of a graph G is a simple graph whose vertices are the maximum matchings of G and where two maximum matchings are adjacent in M（G） if they differ by exactly one edge. In this paper, we prove that if a graph is isomorphic to its maximum matching graph, then every block of the graph is an odd cycle.     

Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

分裂图的区间图完全化问题（英文）

The interval graph completion problem on a graph G is to find an added edge set F such that G + F is an interval supergraph with the smallest possible number of edges. The problem has important applications to numerical algebra, V LSI-layout and algorithm graph theory etc; And it has been known to be N P-complete on general graphs. Some classes of special graphs have been investigated in the literatures. In this paper the interval graph completion problem on split graphs is investigated.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

DIAGNOSABILITY OF CAYLEY GRAPH NETWORKS GENERATED BY TRANSPOSITION TREES UNDER THE COMPARISON DIAGNOSIS MODEL

Diagnosability of a multiprocessor system is one important study topic.Cayley graph network Cay(T_n,S_n) generated by transposition trees Tnis one of the attractive underlying topologies for the multiprocessor system.In this paper,it is proved that diagnosability of Cay(T_n,S_n) is n-1 under the comparison diagnosis model for n ≥ 4.     

Maximum Genus of Strong Embeddings

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover.Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

块图的2-步长控制问题的一个线性时间算法

The 2-step domination problem is to find a minimum vertex set D of a graph such that every vertex of the graph is either in D or at distance two from some vertex of D.In the present paper,by using a labeling method,we provide an O(m) time algorithm to solve the2-step domination problem on block graphs,a superclass of trees.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Genus Distributions of M(o|¨)bius Ladders

The genus distribution of a graph is a polynomial whose coefficients are the partition of the number of embeddings with respect to the genera. In this paper, the genus distribution of Mobius ladders is provided which is an infinite class of 3-connected simple graphs.     

Exploring the constrained maximum edge-weight connected graph problem

Given an edge weighted graph, the maximum edge-weight connected graph （MECG） is a connected subgraph with a given number of edges and the maximal weight sum. Here we study a special case, i.e. the Constrained Maximum Edge-Weight Connected Graph problem （CMECG）, which is an MECG whose candidate subgraphs must include a given set of k edges, then also called the k-CMECG. We formulate the k-CMECG into an integer linear programming model based on the network flow problem. The k-CMECG is proved to be NP-hard. For the special case 1-CMECG, we propose an exact algorithm and a heuristic algorithm respectively. We also propose a heuristic algorithm for the k-CMECG problem. Some simulations have been done to analyze the quality of these algorithms. Moreover, we show that the algorithm for 1-CMECG problem can lead to the solution of the general MECG problem.