On tree congestion of graphs

This paper investigates the problem of embedding a graph into a tree with the same vertex set (a spanning tree in particular), such that the maximum congestion of the edges is minimized. We calculate exact formulas for the tree congestion and spanning tree congestion for various families of graphs, including grids and complete bipartite graphs.     

任意n个完备二分图的并图的k－优美性和算术性

证明了对于正整数k,n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是k-优美图;对于正整数k,d（d≥2）,k≠0（roodd）及n,si,ti（si,ti≥2,i=1,2,…,n）,图n/U/i=1,Ksi,ti是（k,d）-算术图,前一结论推广了文[6]的相应结果。     

The $L(3, 2, 1)$-Labeling on Bipartite Graphs

An $L(3, 2, 1)$-labeling of a graph $G$ is a function from the vertex set $V(G)$ to the set of all nonnegative integers such that $|f(u)−f(v)|≥3$ if $d_G(u, v)=1$, $|f(u)−f(v)|≥2$ if $d_G(u, v)=2$, and $|f(u)−f(v)|≥1$ if $d_G(u, v)=3$. The $L(3, 2, 1)$-labeling problem is to find the smallest number $λ_3(G)$ such that there exists an $L(3, 2, 1)$-labeling function with no label greater than it. This paper studies the problem for bipartite graphs. We obtain some bounds of $λ_3$ for bipartite graphs and its subclasses. Moreover, we provide a best possible condition for a tree $T$ such that $λ_3(T)$ attains the minimum value.     

极大饱和图D（n,k）的极值性质及圈唯一性

本文证明了极大饱和图Ｄ（ｎ，ｋ）的一个极值性质：在与Ｄ（ｎ，ｋ）具有相同度序 所有图中，唯有Ｄ（ｎ，ｋ）含有最少的Ｋ３子图，并由此推出，在几乎正则图的范围内，ｋ个完全图之并及完全ｋ部多分图均是圈唯一的。本文还用图谱方法，证明了完全二分图Ｋｍ，ｎ的圈唯一性。     

On Bipartite and Multipartite Clique Problems

In this paper, we introduce the maximum edge biclique problem in bipartite graphs and the edge/node weighted multipartite clique problem in multipartite graphs. Our motivation for studying these problems came from abstractions of real manufacturing problems in the computer industry and from formal concept analysis. We show that the weighted version and four variants of the unweighted version of the biclique problem are NP-complete. For random bipartite graphs, we show that the size of the maximum balanced biclique is considerably smaller than the size of the maximum edge cardinality biclique, thus highlighting the difference between the two problems. For multipartite graphs, we consider three versions each for the edge and node weighted problems which differ in the structure of the multipartite clique (MPC) required. We show that all the edge weighted versions are NP-complete in general. We also provide a special case in which edge weighted versions are polynomially solvable.     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

ON THE MINIMUM FEASIBLE GRAPH FOR FOUR SETS

ＯＮＴＨＥＭＩＮＩＭＵＭＦＥＡＳＩＢＬＥＧＲＡＰＨＦＯＲＦＯＵＲＳＥＴＳＸＵＹＩＮＦＥＮＧＡＮＤＦＵＸＩＡＯＢＩＮＧＡｂｓｔｒａｃｔ：ＧｉｖｅｎａｃｏｍｐｌｅｔｅｇｒａｐｈｗｉｔｈｖｅｒｔｅｘｓｅｔＸａｎｄｓｕｂｓｅｔｓＸ＿１，Ｘ＿２，．．．，Ｘ＿ｎ...     

Oriented bipartite graphs and the Goldbach graph

In this paper, we study oriented bipartite graphs. In particular, we introduce “bitransitive” graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic bitournaments. As applications, we characterize acyclic bitournaments with Hamiltonian paths, determine the number of non-isomorphic acyclic bitournaments of a given order, and solve the graph-isomorphism problem in linear time for acyclic bitournaments. Next, we prove the well-known Caccetta-Häggkvist Conjecture for oriented bipartite graphs in some cases for which it is unsolved, in general, for oriented graphs. We also introduce the concept of undirected as well as oriented “odd-even” graphs. We characterize bipartite graphs and acyclic oriented bipartite graphs in terms of them. In fact, we show that any bipartite graph (acyclic oriented bipartite graph) can be represented by some odd-even graph (oriented odd-even graph). We obtain some conditions for connectedness of odd-even graphs. This study of odd-even graphs and their connectedness is motivated by a special family of odd-even graphs which we call “Goldbach graphs”. We show that the famous Goldbach's conjecture is equivalent to the connectedness of Goldbach graphs. Several other number theoretic conjectures (e.g., the twin prime conjecture) are related to various parameters of Goldbach graphs, motivating us to study the nature of vertex-degrees and independent sets of these graphs. Finally, we observe Hamiltonian properties of some odd-even graphs related to Goldbach graphs for a small number of vertices.     

The bipartite edge frustration of composite graphs

The smallest number of edges that have to be deleted from a graph to obtain a bipartite spanning subgraph is called the bipartite edge frustration of G and denoted by φ(G). In this paper we determine the bipartite edge frustration of some classes of composite graphs.     

The complexity of the matching‐cut problem for planar graphs and other graph classes

The Matching‐Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be ‐complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains ‐complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is ‐complete for planar graphs with girth five. The reduction is from planar graph 3‐colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching‐cuts are described. These classes include claw‐free graphs, co‐graphs, and graphs with fixed bounded tree‐width or clique‐width. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 109–126, 2009     

Minimum edge ranking spanning trees of split graphs

Given a graph G, the minimum edge ranking spanning tree problem (MERST) is to find a spanning tree of G whose edge ranking is minimum. However, this problem is known to be NP-hard for general graphs. In this paper, we show that the problem MERST has a polynomial time algorithm for split graphs, which have useful applications in practice. The result is also significant in the sense that this is a first non-trivial graph class for which the problem MERST is found to be polynomially solvable. We also show that the problem MERST for threshold graphs can be solved in linear time, where threshold graphs are known to be split.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

关于k-消去二分图的一些结果

图 G的一个 k-正则支撑子图称为 G的 k-因子 ,若对 G的任一边 e,图 G- e总存在一个 k-因子 ,则称 G是 k-消去图 .证明了二分图 G=( X,Y) ,且 | X | =| Y|是 k-消去图的充分必要条件是 k| S|≤ r1 + 2 r2 +…+ k( rk+… + rΔ) - ε( S)对所有 S X成立 .并由此给出二分图是 k-消去图的充分度条件 .     

Minimum congestion spanning trees in planar graphs

The main purpose of the paper is to develop an approach to the evaluation or the estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.     

On the strong chromatic index and maximum induced matching of tree-cographs,permutation graphs and chordal bipartite graphs

We show that there exist linear-time algorithms that compute the strong chromatic index and a maximum induced matching of tree-cographs when the decomposition tree is a part of the input. We also show that there exist efficient algorithms for the strong chromatic index of (bipartite) permutation graphs and of chordal bipartite graphs.     

几乎无爪图的叶子数较少的生成树

A spanning tree with no more than 3 leaves is called a spanning 3-ended tree.In this paper, we prove that if G is a k-connected(k ≥ 2) almost claw-free graph of order n and σ_(k+3)(G) ≥ n + k + 2, then G contains a spanning 3-ended tree, where σk(G) =min{∑_(v∈S)deg(v) : S is an independent set of G with |S| = k}.     

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.     

图Cn及其r-冠的新的优美标号

研究了关于图的r-冠的优美标号的一个问题,证明了：当n≡0,3（mod 4）时,图Cn及其r-冠是优美图,所给出的新的优美标号不同于现有文献中得到的结果.进而证明了当n≡0（mod 4）时,图Cn及其r-冠也是交错图.     

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v∈V(G). The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.     

关于给定控制数的连通二部图的极大图的刻画

本文研究了给定控制数的连通二部图的极大图的结构问题.利用分类讨论思想和数学归纳法,刻画了控制数等于3和大于等于4这两类边数达到极值时的连通二部图.本文所得结果可用于进一步研究给定全控制数的连通二部图的极大图问题.