Toughness, hamiltonicity and split graphs

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.

2018

In this paper we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of K1,4-free split graphs in the context of Hamiltonian path problem, and as a consequence, we obtain a polynomial-time algorithm to the Hamiltonian path problem in K1,4-free split graphs. We close this paper with the hardness result: we show that, unless P=NP, Hamiltonian path problem is NP-complete in K1,5-free split graphs by reducing from Hamiltonian cycle problem in K1,5-free split graphs. Thus this paper establishes a “thin complexity line” separating NP-complete instances and polynomial-time solvable instances.

Let t # 1 be an integer. We show that it is NP-hard to determine if an r-regular graph is t-tough for any fixed integer r # 3 t.Wealso discuss the complexity of recognizing if an r-regular graph is t-tough, for any rational t # 1. Keywords : toughness, regular graph, NP-completeness AMS Subject Classifications (1991) : 68R10, 05C38 # Supported in part by the National Science Foundation under Grant DMS9206991. 1 1 Introduction We begin with a few definitions and some notation. A good reference for any undefined terms in graph theory is [7] and for computational complexity is [12]. We consider only undirected graphs with no loops or multiple edges. Let G be a graph. Then G is hamiltonian if it has a Hamilton cycle, i.e., a cycle containing all of its vertices. It is traceable if it has a path containing all of its vertices. Let #(G) denote the number of components of G.We say G is t-tough if |S|#t #(G - S) for every subset S of the vertex set V (G)ofG with #(G - S) > 1...

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.

A graph G=(V,E) is called a split graph if there exists a partition V=I∪K such that the subgraphs G[I] and G[K] of G induced by I and K are empty and complete graphs, respectively. In this paper, we survey results on the hamiltonian and classification problems for split graphs G with the minimum degree δ(G)≥|I|-4.

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.

2020

LetG be a graph of order n ≥ 6r−2, size e, and minimum degree δ ≥ r, where r is an integer greater than 1. The main result obtained in this note is that ifG is 1-tough with the degree sequence (d1, d2, · · · , dn) and if e ≥ ((n−r)(n−r−1)+r(2r−1))/2, then G is pancyclic, Hamiltonian bipartite, or Hamiltonian such that d1 = d2 = · · · = dk = k, dk+1 = dk+2 = · · · = dn−k+1 = n − k − 1, and dn−k+2 = dn−k+3 = · · · = dn = n − 1, where k < n/2. This result implies that the following conjecture of Hoa, posed in 2002, is true under the conditions n ≥ 40 and δ ≥ 7: every path-tough graph on n vertices and with at least ((n− 6)(n− 7)+ 34)/2 edges is Hamiltonian. Using the main result of this note, additional sufficient conditions for 1-tough graphs to be pancyclic are also obtained.

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.

Discrete Mathematics, 1997

We show that every 1-tough cocomparability graph has a Hamilton cycle. This settles a conjecture of Chvfital 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.

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.