Hamilton cycles in 6-connected claw-free graphs (Extended abstract)

Thomassen conjectured in 1986 that every 4-connected line graph is hamiltonian. In this paper, we show that 6-connected line graphs are hamiltonian, improving on an analogous result for 7-connected line graphs due to Zhan in 1991. Our result implies that every 6-connected claw-free graph is hamiltonian.     

Almost all Cayley graphs are hamiltonian

It has been conjectured that there is a hamiltonian cycle in every finite connected Cayley graph. In spite of the difficulty in proving this conjecture, we show that almost all Cayley graphs are hamiltonian. That is, as the order n of a groupG approaches infinity, the ratio of the number of hamiltonian Cayley graphs ofG to the total number of Cayley graphs ofG approaches 1.Supported by the National Natural Science Foundation of China, Xinjiang Educational Committee and Xinjiang University.     

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

In 1956, W.T. Tutte proved that every 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. It is shown that Sanders’ result is best possible by constructing 4-connected maximal planar graphs with three edges a large distance apart such that any hamiltonian cycle misses one of them. If the maximal planar graph is 5-connected then such a construction is impossible.     

中国投递员问题综述

§1 引言 中国投递员问题(Chinese Postman Problem)是1960年我们从生产实际中提出的一个数学问题,它是从下述实际问题中抽象出来的:“一个投递员应该怎样选择一条路线,才能既把所有由他负责送的信都送到,而所走的路程又最短。” 在我们开始研究中国投递员问题以前,国外有人研究过所谓旅行售货员问题     

3‐colorability of 4‐regular hamiltonian graphs

On the model of the cycle‐plus‐triangles theorem, we consider the problem of 3‐colorability of those 4‐regular hamiltonian graphs for which the components of the edge‐complement of a given hamiltonian cycle are non‐selfcrossing cycles of constant length ≥ 4. We show that this problem is NP‐complete. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 125–140, 2003     

Series parallel extensions of plane graphs to dual-eulerian graphs

A plane graph is dual-eulerian if it has an eulerian tour with the property that the same sequence of edges also forms an eulerian tour in the dual graph. Dual-eulerian graphs are of interest in the design of CMOS VLSI circuits.Every dual-eulerian plane graph also has an eulerian Petrie (left–right) tour thus we consider series-parallel extensions of plane graphs to graphs, which have eulerian Petrie tours. We reduce several special cases of extensions to the problem of finding hamiltonian cycles. In particular, a 2-connected plane graph G has a single series parallel extension to a graph with an eulerian Petrie tour if and only if its medial graph has a hamiltonian cycle.     

Independent dominating sets and hamiltonian cycles

A graph is uniquely hamiltonian if it contains exactly one hamiltonian cycle. In this note we prove that there are no r‐regular uniquely hamiltonian graphs when r > 22. This improves upon earlier results of Thomassen. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 233–244, 2007     

Updating the hamiltonian problem—A survey

This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area.     

A census of maximum uniquely hamiltonian graphs

We show that there are 2^[n/2]-4 largest graphs of order n ≥ 7 having exactly one hamiltonian cycle. a recursive procedure for constructing these graphs is described.     

Hamiltonian properties of triangular grid graphs

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly-convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearly-convex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph D) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable. We also show that the problem Hamiltonian Cycle is NP-complete for triangular grid graphs.     

Some recent results in hamiltonian graphs

A variety of recent developments in hamiltonian theory are reviewed. In particular, several sufficient conditions for a graph to be hamiltonian, certain hamiltonian properties of line graphs, and various hamiltonian properties of powers of graphs are discussed. Furthermore, the concept of an n-distant hamiltonian graph is introduced and several theorems involving this special class of hamiltonian graphs are presented.     

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

A number of results in hamiltonian graph theory are of the form “ implies ”, where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. An example of such a theorem is the well-known Chvátal–Erd s Theorem, which states that every graph G with κ is hamiltonian. Here κ is the vertex connectivity of G and is the cardinality of a largest set of independent vertices of G. In another paper Chvátal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than κ independent vertices. In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of a well-known theorem of Jung (Ann. Discrete Math. 3 (1978) 129) for graphs on 16 or more vertices.     

Cycles in 2-Factors of Balanced Bipartite Graphs

In the study of hamiltonian graphs, many well known results use degree conditions to ensure sufficient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with . Revised: May 7, 1999     

Chromatic properties of hamiltonian graphs

We study the problem of the location of real zeros of chromatic polynomials for some families of graphs. In particular, a problem presented by Thomassen (see [On the number of hamiltonian cycles in bipartite graphs, Combin. Probab. Comput. 5 (1996) 437–442.]) is discussed and a result for hamiltonian graphs is presented. An open problem is stated for 2-connected graphs with a hamiltonian path.     

3-连通、高次和坚韧图周长的估计(Ⅰ)

设G是一个n阶3-连通图,周长为C(G),独立数为,若G是1-坚韧的,且,则G的每一个最长圈是控制圈且;又若G是5/3-坚韧的或,则G是Hamilton图。     

Tough spiders

Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2‐tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/2 – ε)‐tough spider graphs that do not contain a Hamilton cycle. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 23–40, 2007     

Longest paths and cycles in K1,3-free graphs

In this article we show that the standard results concerning longest paths and cycles in graphs can be improved for K_1,3-free graphs. We obtain as a consequence of these results conditions for the existence of a hamiltonian path and cycle in K_1,3-free graphs.     

Schrijver图SG(2k+2,k)的Hamilton性

通过图G的每个顶点的路称为Hamilton路,通过图G的每个顶点的圈称为Hamilton圈,具有Hamilton圈的图G称为Hamilton图.1952年Dirac曾得到关于Hamilton图一个充分条件的结论:图G有n个顶点,如果每个顶点υ满足:d(υ)≥n/2,则图G是Hamilton图.本文研究了Schrijver图SG(2k+2,k)的Hamilton性,采用寻找Hamilton圈的方法得出了Schrijver图SG(2k+2,k)是Hamilton图.     

Old Hamiltonian Ideas from a New Point of View

We prove a sufficient condition for graphs to be hamiltonian. This result generalizes five sufficient conditions for hamiltonian graphs and is non-comparable with many well-known ones.     

Independent Production of Non Hamiltonian Graphs

The decision whether a graph is hamiltonian or not is known to be an NP-complete problem. The importance of this kind of problem motivate several researchers in heuristics development. However, problems arise in the evaluating of this heuristics, more often because it is difficult to produce independent data. In this paper we develop methods to produce non hamiltonian graphs, based on independence subsets and toughness arguments. We also present a family of non hamiltonian graphs with strong restrictions, that is, planar 1-tough non hamiltonan graphs with no separation triangles.