Toughness, degrees and 2-factors

Journal of Graph Theory, 1996

Let G be a planar graph on n vertices, let c(G) denote the length of a longest cycle of G, and let w(G) denote the number of components of G. By a well-known theorem of Tutte, c(G) = n (i.e., G is hamiltonian) if G is 4-connected. Recently, Jackson and Wormald showed that c(G) 2 ona for some positive constants , B and cy M 0.2 if G is 3-connected. Now let G have connectivity 2. Then c(G) may be as small as 4, as with K2,n-21 unless we bound w(G-S) for every subset S of V(G) with IS1 = 2. Define t(G) as the maximum of w(G-S) taken over all 2-element subsets S C V(G). We give an asymptotically sharp lower bound for the toughness of G in terms of J(G), and we show that c(G) 2 81nn for some positive constant 8 depending only on E(G). In the proof we use a recent result of Gao and Yu improving Jackson and Wormald's result. Examples show that the lower bound on c(G) is essentially best-possible. o 1996 John Wiley & Sons, Inc.

Graphs and Combinatorics, 1999

A graph is rY 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 ttough graph of order n with t 2 is 2Y k-factor-critical for every non-negative integer k minf2t À 2Y n À 3g, thus proving a conjecture as well as generalizing the main result of Liu and Yu in .

Graphs Comb., 2021

The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1\square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_n\square H$ is Hamiltonian; (ii) $P_n\square H$ is $1$-tough; and (iii) $H$ has a path factor.

Graphs and Combinatorics, 2006

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! 2 Douglas Bauer et al.

Graphs and Combinatorics, 1999

A graph is rY k-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. an r-regular spanning subgraph). Let tG denote the toughness of graph G. In this paper, we show that if tG 4, then G is 3Y k-factor-critical for every non-negative integer k such that n k even, k`2tG À 2 and k n À 7.

Electronic Notes in Discrete Mathematics, 2002

We now know that not every 2-tough graph is hamiltonian. In fact for every > 0, there exists a (9/4− ) -tough nontraceable graph. We continue our quadrennial survey of results that relate the toughness of a graph to its cycle structure.

Discrete Mathematics, 2002

A graph G is called triangle-free if G has no induced K3 as a subgraph. We set 3 = min{ 3 i=1 d(vi)|{v1; v2; v3} is an independent set of vertices in G}. In this paper, we show that if G is a 1-tough and triangle-free graph of order n with n 6 3, then G is hamiltonian.

Iconic Research and Engineering Journals, 2019

In this paper we mention non separable components of longest cycles. And then we establish chordality and 2-factor in tough graph. A graph G is chordal if it contains no chordless cycle of length at least four and is k-chordal if a longest chordless cycle in G has length at most k. Finally the result reveals that all 3/2-tough 5-chordal graph G with a 2-factor are obtained.

Journal of Graph Theory, 2013

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t-tough. We first give a best monotone theorem when t ≥ 1, but then show that for any integer k ≥ 1, a best monotone theorem for t = 1 k ≤ 1 requires at least f (k) · |V (G)| nonredundant conditions, where f (k) grows superpolynomially as k → ∞. When t < 1, we give two simple theorems for G to be t-tough, in terms of its vertex degrees. We conclude with a theorem giving the best possible minimum degree condition for G to be t-tough,

Discrete Applied Mathematics

The prism over a graph G is the product G K 2 , i.e., the graph obtained by taking two copies of G and adding a perfect matching joining the two copies of each vertex by an edge. The graph G is called prism-hamiltonian if it has a hamiltonian prism. Jung showed that every 1-tough P 4-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for k ≥ 1 a P 4-free graph has a spanning kwalk (closed walk using each vertex at most k times) if and only if it is 1 k-tough. As our main result, we show that for the class of P 4-free graphs, the three properties of being prism-hamiltonian, having a spanning 2-walk, and being 1 2-tough are all equivalent.