Tough graphs and hamiltonian circuits

The toughness of a graph G is defined as the largest real number t such that deletion of any s points from G results in a graph which is either connected or else has at most s/t components. Clearly, every hamiltonian graph is 1-tough. Conversely, we conjecture that for some t₀, every t₀-tough graph is hamiltonian. Since a square of a k-connected graph is always k-tough, a proof of this conjecture with t₀ = 2 would imply Fleischner's theorem (the square of a block is hamiltonian). We construct an infinite family of ($\text{3}\text{2}$)-tough nonhamiltonian graphs.     

Toughness, hamiltonicity and split graphs

Related to Chvátal's famous conjecture stating that every 2-tough graph is hamiltonian, we study the relation of toughness and hamiltonicity on special classes of graphs.

First, we consider properties of graph classes related to hamiltonicity, traceability and toughness concepts and display some algorithmic consequences. Furthermore, we present a polynomial time algorithm deciding whether the toughness of a given split graph is less than one and show that deciding whether the toughness of a bipartite graph is exactly one is coNP-complete.

We show that every split graph is hamiltonian and that there is a sequence of non-hamiltonian split graphs with toughness converging to .     

Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G−S has a 1-factor for any SV(G) with |S|=n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with , then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G−S has a k-factor for any SV(G) with |S|=n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with , then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k2.     

1-Tough cocomparability graphs are hamiltonian

We show that every 1-tough cocomparability graph has a Hamilton cycle. This settles a conjecture of Chvàtal for the case of cocomparability graphs. Our approach is based on exploiting the close relationship of the problem to the scattering number and the path cover number.     

Properties of Edge-Tough Graphs

 In [6] the author defined a new property of graphs namely the edge-toughness. It was proved in [6] that a 2t-tough graph is always t-edge-tough. It is proved in the present paper that this is not true for (2t−ε)-tough graphs if t is a positive integer. A result of Enomoto et al. in [5] implies that every 2-tough graph has a 2-factor. In the present paper it is proved that every 1-edge-tough graph has a 2-factor. This is a sharpening of the previous statement. Received: July 10, 1996 Revised: March 4, 1998     

Nonseparable graphs with a given number of cycles

G. Ringel conjectured that for every positive integer n other than 2, 4, 5, 8, 9, and 16, there exists a nonseparable graph with n cycles. It is proved here that the conjecture is true even with the restriction to planar and hamiltonian graphs.     

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.     

A Remark on Hamiltonian Cycles

Let G be an undirected and simple graph on n vertices. Let ω, α and χ denote the number of components, the independence number and the connectivity number of G. G is called a 1-tough graph if ω(G – S) ? |S| for any subset S of V(G) such that ω(G ? S) > 1. Let σ₂ = min {d(v) + d(w)|v and w are nonadjacent}. Note that the difference α - χ in 1-tough graph may be made arbitrary large. In this paper we prove that any 1-tough graph with σ₂ > n + χ - α 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.     

Thomassen's conjecture implies polynomiality of 1‐Hamilton‐connectedness in line graphs

A graph G is 1‐Hamilton‐connected if G?x is Hamilton‐connected for every x∈V(G), and G is 2‐edge‐Hamilton‐connected if the graph G+ X has a hamiltonian cycle containing all edges of X for any X?E⁺(G) = {xy| x, y∈V(G)} with 1≤|X|≤2. We prove that Thomassen's conjecture (every 4‐connected line graph is hamiltonian, or, equivalently, every snark has a dominating cycle) is equivalent to the statements that every 4‐connected line graph is 1‐Hamilton‐connected and/or 2‐edge‐Hamilton‐connected. As a corollary, we obtain that Thomassen's conjecture implies polynomiality of both 1‐Hamilton‐connectedness and 2‐edge‐Hamilton‐connectedness in line graphs. Consequently, proving that 1‐Hamilton‐connectedness is NP‐complete in line graphs would disprove Thomassen's conjecture, unless P = NP. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 241–250, 2012     

Tough Ramsey Graphs Without Short Cycles

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a conjecture of Chvátal that there exists an absolute constant t ₀ so that every t ₀-tough graph is pancyclic. The proof is by an explicit construction based on the tight relationship between the spectral properties of a regular graph and its expansion properties. A similar technique provides a simple construction of triangle-free graphs with independence number m on (m ^4/3) vertices, improving previously known explicit constructions by Erdös and by Chung, Cleve and Dagum.     

(2,k)-Factor-Critical Graphs and Toughness

 A graph is (r,k)-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. with an r-regular spanning subgraph). We show that every τ-tough graph of order n with τ≥2 is (2,k)-factor-critical for every non-negative integer k≤min{2τ−2, n−3}, thus proving a conjecture as well as generalizing the main result of Liu and Yu in [4]. Received: December 16, 1996 / Revised: September 17, 1997     

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.     

HAMILTONIAN DECOMPOSITION OF CAYLEY GRAPHS OF ORDERS p~2 AND pq

1. IntroductionLet G be a finite group and S a subset of G such that S--1 ~ S, and 1 f S. The Cayleygraph Cay (G, S) is defined as the simple graph with V ~ G, and E = {glgZ I g,'g, or g,'g,6 S, gi, gi E G}. Cay (G, S) is vertex-transitive, and it is connected if and only if (S) = G,i.e. S is a generating set of G[1]. If G = Zn, then Cay (Zn, S) is called a circulant graph. Ithas been proved that any connected Cayley graph on a finite abelian group is hamiltonianl2].Furthermore, …     

Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

Two sufficient conditions for a 2-factor in a bipartite graph

In this paper we prove that every 1-tough bipartite graph which is not isomorphic to K_1,1 has a 2-factor. We also obtain a sufficient condition for the existence of a 2-factor in a bipartite graph, in the spirit of Hall's theorem.     

Hourglasses and Hamilton cycles in 4‐connected claw‐free graphs

We show that if G is a 4‐connected claw‐free graph in which every induced hourglass subgraph S contains two non‐adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4‐connected claw‐free, hourglass‐free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005     

2-factors in triangle-free graphs

We study the cycle structure of 1-tough, triangle-free graphs. In particular, we prove that every such graph on n ≥ 3 vertices with minimum degree δ ≥ 1/4 (n + 2) has a 2-factor. We also show this is best possible by exhibiting an infinite class of 1-tough, triangle-free graphs having δ = 1/4(n + 1) and no 2-factor. © 1996 John Wiley & Sons, Inc.     

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.     

Toughness in Graphs – A Survey

In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!