图的树宽的分解定理

图的树宽问题是名的ＮＰ－困难问题。其分解原则在确定树宽的一般算法和特殊算法中有重要应用。本给出这方面的若干定理。     

两个完全图的乘积的树宽

本文确定了乘积图Km×Kn的树宽.我们的结果是若m和n都是偶数,且m≥n,或m是奇数而n是偶数,或m和n都是奇数且n≥m,则Km×Kn的树宽是TW(Km×Kn)=n(m+1)/2-1.这恰好是图Km×Kn的带宽.     

平面可弦图的子式障碍

证明了平面可弦图的子式障碍恰由Ｋ５，Ｋ３，３，Ｋ２，２，２和Ｋ２×Ｃ５这四个图构成     

关于无爪Hamilton图的一个泛圈性结果

一个图Ｇ是泛圈的，如果它含有长为３，４，…，ｎ（＝｜Ｖ（Ｇ）｜）的圈．本文探讨了一类无爪Ｈａｍｉｌｔｏｎ图的圈结构，主要结果为：设Ｇ＝（Ｖ，Ｅ）是ｎ阶无爪Ｈａｍｉｌｔｏｎ图．如果Ｇ中有节点ｘ使ｄ（ｘ）≧ｎ／２且Ｎ（ｘ）连通，则除少数几个例外，Ｇ是泛圈的．     

图的生成树分解

一个图Ｇ称为是ｍ－ＳＴ可分解的,如果Ｇ能分解为ｍ个边不交生成树的并,本文研究了一个图是ｍ－ＳＴ可分解的若干性质,并证明了两类平面图是２－ＳＴ可分解的。     

舵图的优美性

在“东南亚第十一届组合图论与计算机学术会议”上,K.M.Koh与D.G.Rogers等人提出了一个猜想:“对所有n≥3,舵图H_n为优美图”。本文证明了这一猜想为真。     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

树与偏k—的乘积的树宽

本文确宇了一棵树与一个k-连通偏k-树的乘积图的树宽。其中,偏k-树是一个树宽为k的图。     

无K_4-图子式的图的谱半径(英文)

Ｇ是一个无Ｋ４－图子式、顶点数为ｎ的简单图,ｐ（Ｇ）是图Ｇ的谱半径．本文得出一个关于ｐ（Ｇ）的上确界：等式成立当且仅当 Ｇ ≌Ｋ２　（ｎ－２）Ｋ１,其中 Ｇ１　Ｇ２是由 Ｇ１∪Ｇ２组成、并且Ｇ１中的第一个点和Ｇ２中的每一个点之间都有一条边相连：（ｎ－２）Ｋ１表示（ｎ－２）个孤立点的集合．     

关于图的Hamilton性

凡未作解释的术语均可参考Bondy和Murty的书。 一个图G=(V,E),如果满足如下的性质A和B,则称之为核心图。所有核心图的集合记为。 性质A存在一个整数K≥1使得:(i)V=V_o V_1 … V_k;(ii)G[V-V_o)=G[V_1)     

Treewidth of Cartesian Products of Highly Connected Graphs

The following theorem is proved: for all k‐connected graphs G and H each with at least n vertices, the treewidth of the cartesian product of G and H is at least . For , this lower bound is asymptotically tight for particular graphs G and H. This theorem generalizes a well‐known result about the treewidth of planar grid graphs.     

Parameters Tied to Treewidth

Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory. This article surveys several graph parameters tied to treewidth, including separation number, tangle number, well‐linked number, and Cartesian tree product number. We review many results in the literature showing these parameters are tied to treewidth. In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

Topological Minors of Cover Graphs and Dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that ‐free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.     

Hadwiger's Conjecture for the Complements of Kneser Graphs

Hadwiger's conjecture asserts that every graph with chromatic number t contains a complete minor of order t. Given integers , the Kneser graph is the graph with vertices the k‐subsets of an n‐set such that two vertices are adjacent if and only if the corresponding k‐subsets are disjoint. We prove that Hadwiger's conjecture is true for the complements of Kneser graphs.     

k-树的补图的最小填充和树宽

一个图的最小填充问题是寻求边数最少的弦母图,一个图的树宽问题是寻求团数最小的弦母图,这两个问题分别在稀疏矩阵计算及图的算法设计中有非常重要的作用．一个k-树G的补图G称为k-补树．本文给出了k-补树G的最小填充数f(G) 及树宽TW(G)．     

Hall‐Type Results for 3‐Connected Projective Graphs

Projective planar graphs can be characterized by a set of 35 excluded minors. However, these 35 are not equally important. A set of 3‐connected members of is excludable if there are only finitely many 3‐connected nonprojective planar graphs that do not contain any graph in as a minor. In this article, we show that there are precisely two minimal excludable sets, which have sizes 19 and 20, respectively.     

A bound on the treewidth of planar even-hole-free graphs

The class of planar graphs has unbounded treewidth, since the k×k grid, ∀k∈N, is planar and has treewidth k. So, it is of interest to determine subclasses of planar graphs which have bounded treewidth. In this paper, we show that if G is an even-hole-free planar graph, then it does not contain a 9×9 grid minor. As a result, we have that even-hole-free planar graphs have treewidth at most 49.