Color-critical graphs and hypergraphs

The main purpose of this paper is to present a technique for obtaining constructions of color-critical graphs. The technique consists in reducing color-critical hypergraphs to color-critical graphs, and the constructions obtained generalize and unify known constructions. It is suggested that constructions of other classes of graphs may be obtained by a similar technique.     

线团-收敛图

一个图的线团图就是这个图的线图的团图。对于自然数n，一个图被称为n-线团－收敛的，如果它的n次线团图同构于一个固定的图。否则称之为发散的。本刻画了线团－收敛图与发散图，给出一个线团－收敛图的构造方法，并且，讨论了线团－收敛图的线团－收敛指数。     

三圈图的Harary指数

图G的Harary指数是指图G中所有顶点对间的距离倒数之和. 三圈图是指边数等于顶点数加2的连通图. 研究了三圈图的Harary数, 给出了所有三圈图中具有极大Harary指数的图的结构以及含有三个圈的三圈图中具有次大Harary指数的图的结构.     

Lexicographic orientation and representation algorithms for comparability graphs,proper circular arc graphs,and proper interval graphs

We introduce a simple new technique which allows us to solve several problems that can be formulated as seeking a suitable orientation of a given undirected graph. In particular, we use this technique to recognize and transitively orient comparability graphs, to recognize and represent proper circular arc graphs, and to recognize and represent proper interval graphs. As a consequence, we derive and represent proper interval graphs. As a consequence, we derive simple new proofs of a theorem of Ghouila-Houri and a theorem of Skrien. Our algorithms are conceptually simpler than (and often of comparable efficiency to) the existing algorithms for these problems. © 1995 John Wiley & Sons, Inc.     

Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.     

Calculating the edge Wiener and edge Szeged indices of graphs

The edge Szeged and edge Wiener indices of graphs are new topological indices presented very recently. It is not difficult to apply a modification of the well-known cut method to compute the edge Szeged and edge Wiener indices of hexagonal systems. The aim of this paper is to propose a method for computing these indices for general graphs under some additional assumptions.     

分子图的Zagreb拓扑指标的逆问题

研究了化学分子图的Zagreb指标的逆问题，解决了对于给定的怎样的数存在分子图，其Zagreb指标值等于该数的问题，对n个顶点m条边的简单连通图，给出了其具有最小Zagreb指标值的充分必要条件，并给出了其具有最大Zagreb指标值的必要条件，为利用计算机搜索具有给定Zagreb指标值的所有分子图界定了顶点数和边数的范围，从而提高了计算机搜索的效率，这在组合化学中具有重要的意义。     

三圈图的极小广义和连通指数

图的广义和连通指数作为新提出的一类分子拓扑指数, 在QSPR/QSAR 中有很大的应用价值. 树图、单圈图和双圈图的极值问题已取得很多结果, 而三圈图相关问题的研究较为复杂. 限制 - 1\leqslant \alpha < 0, 对三圈图的广义和连通指数进行了研究. 通过对三圈图的分析, 构造了一种图的变换, 指出在三圈图中广义和连通指 数的极小值必由其中的七种类型图取得. 然后通过悬挂边的变换, 最终得到三圈图广义和连通指 数的极小值并刻画了唯一的极图.     

Weight choosability of graphs

Suppose the edges of a graph G are assigned 3‐element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different? We prove that the answer is positive for several classes of graphs, including complete graphs, complete bipartite graphs, and trees (except K₂). The argument is algebraic and uses permanents of matrices and Combinatorial Nullstellensatz. We also consider a directed version of the problem. We prove by an elementary argument that for digraphs the answer to the above question is positive even with lists of size two. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 242–256, 2009     

On cubical graphs

It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of n-dimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs G, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an n-dimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgraphs can be embedded.     

Tricyclic graphs with maximum Merrifield-Simmons index

It is well known that the graph invariant, ‘the Merrifield-Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield-Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16-21]. The sharp upper and lower bounds for the Merrifield-Simmons indices of unicyclic graphs are characterized by Pedersen and Vestergaard [A.S. Pedersen, P.D. Vestergaard, The number of independent sets in unicyclic graphs, Discrete Appl. Math. 152 (2005) 246-256]. The sharp upper bound for the Merrifield-Simmons index of bicyclic graphs is obtained by Deng, Chen and Zhang [H. Deng, S. Chen, J. Zhang, The Merrifield-Simmons index in (n,n+1)-graphs, J. Math. Chem. 43 (1) (2008) 75-91]. The sharp lower bound for the Merrifield-Simmons index of bicyclic graphs is determined by Jing and Li [W. Jing, S. Li, The number of independent sets in bicyclic graphs, Ars Combin, 2008 (in press)]. In this paper, we will consider the tricyclic graph, i.e., a connected graph with cyclomatic number 3. The tricyclic graph with n vertices having maximum Merrifield-Simmons index is determined.     

On the Hosoya index of graphs

Let G be a (molecular) graph. The Hosoya index Z(G) of G is defined as the number of subsets of the edge set E(G) in which no two edges are adjacent in G, i.e., Z(G) is the total number of matchings of G. In this paper, we determine all the connected graphs G with n + 1 ≤ Z(G) ≤ 5n ? 17 for n ≥ 19. As a byproduct, the graphs of n vertices with Hosoya index from the second smallest value to the twenty first smallest value are obtained for n ≥ 19.     

Universal graphs and induced-universal graphs

We construct graphs that contain all bounded-degree trees on n vertices as induced subgraphs and have only cn edges for some constant c depending only on the maximum degree. In general, we consider the problem of determining the graphs, so-called universal graphs (or induced-universal graphs), with as few vertices and edges as possible having the property that all graphs in a specified family are contained as subgraphs (or induced subgraphs). We obtain bounds for the size of universal and induced-universal graphs for many classes of graphs such as trees and planar graphs. These bounds are obtained by establishing relationships between the universal graphs and the induced-universal graphs.     

Improved algorithms for replacement paths problems in restricted graphs

We present near-optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. Our technique improves the existing bounds for these problems on directed acyclic graphs, planar graphs, and non-planar integer-edge-weighted graphs.     

Cover-incomparability graphs and chordal graphs

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxová, P. Pavlíkova, A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.     

Colored graphs and matrix integrals

We discuss applications of generating functions for colored graphs to asymptotic expansions of matrix integrals. The described technique provides an asymptotic expansion of the Kontsevich integral. We prove that this expansion is a refinement of the Kontsevich expansion, which is the sum over the set of classes of isomorphic ribbon graphs. This yields a proof of Kontsevich’s results that is independent of the Feynman graph technique. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 8–24.     

Dominating sets in social network graphs

This paper presents an approach to the study of the structure of social networks using dominating sets of graphs. A method is presented for partitioning the vertices of a graph using dominating vertices. For certain classes of graphs this is helpful in determining the underlying structure of the corresponding social network. An extension of this technique provides a method of studying the structure of directed graphs and directed social networks. Minimum dominating sets are related to statuses and structurally equivalent sets.     

A framework and algorithms for circular drawings of graphs

In this paper, we present a framework and two linear time algorithms for obtaining circular drawings of graphs. The first technique produces circular drawings of biconnected graphs and finds a zero crossing circular drawing if one exists. The second technique finds multiple embedding circle drawings. Techniques for the reduction of edge crossings are also discussed. Results of experimental studies are included.     

Anticodes for the Grassman and bilinear forms graphs

In [2], L. Chihara proved that many infinite families of classical distance-regular graphs have no nontrivial perfect codes, including the Grassman graphs and the bilinear forms graphs. Here, we present a new proof of her result for these two families using Delsarte's anticode condition[3]. The technique is an extension of an approach taken by C. Roos [6] in the study of perfect codes in the Johnson graphs.