Maximal Nontraceable Graphs with Toughness less than One

Opuscula Mathematica, 2009

By NT(n) we denote the set of graphs of order n which are traceable but have non-traceable edges, i.e. edges which are not contained in any hamiltonian path. The class NT(n) has been considered by Balińska and co-authors in a paper published in 2003, where it was proved that the maximum size tmax(n) of a graph in NT(n) is at least (n 2 − 5n + 14)/2 (for n ≥ 12). The authors also found tmax(n) for 5 ≤ n ≤ 11. We prove that, for n ≥ 5, tmax(n) = max{`n −2 2´+ 4,`n − n−1 2 2´+ n−1 2 2 } and, moreover, we characterize the extremal graphs (in fact we prove that these graphs are exactly those already described in the paper by Balińska et al.). We also prove that a traceable graph of order n ≥ 5 may have at most n−3 2 n−3 2 non traceable edges (this result was conjectured in the mentioned paper by Balińska and co-authors).

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, 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.

1999

A path of a graph is maximal if it is not a proper subpath of any other path of the graph. A graph is scenic if every maximal path of the graph is a maximum length path. In [4] we give a new proof of C. Thomassen's result characterizing all scenic graphs with Hamiltonian path. Using similar methods here we determine all scenic graphs with no Hamiltonian path.

Some problems related to security in communication networks lead to consider a new type of connectivity in graphs, namely the confidential connectivity. In this paper we present a characterization of unbreakable graphs using the notion of weak decomposition and we give some applications of minimal unbreakable graphs. In fact, we showed that a graph G is confidentially connected if and only if it does not have a star cutset. We also showed that a minimal imperfect graph does not have a star cutset. We gave a constructive proof of the fact that every (α, ω)-partitionable graph is confidentially connected, for a superclass of minimal imperfect graphs.

Involve, a Journal of Mathematics, 2017

The problem of characterizing maximal non-Hamiltonian graphs may be naturally extended to characterizing graphs that are maximal with respect to non-traceability and beyond that to t-path traceability. We show how traceability behaves with respect to disjoint union of graphs and the join with a complete graph. Our main result is a decomposition theorem that reduces the problem of characterizing maximal t-path traceable graphs to characterizing those that have no universal vertex. We generalize a construction of maximal non-traceable graphs by Zelinka to t-path traceable graphs.

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.

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.

Journal of the Operations Research Society of China, 2014

Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters, such as connectivity, toughness, scattering number, integrity, tenacity, rupture degree and edge-analogues of some of them. Among these parameters, the tenacity and rupture degree are two better ones to measure the stability of a network. In this paper we consider two extremal problems on the tenacity of graphs: Determine the minimum and maximum tenacity of graphs with given order and size. We give a complete solution to the first problem, while for the second one, it turns out that the problem is much more complicated than that of the minimum case. We determine the maximum tenacity of trees and unicyclic graphs with given order and show the corresponding extremal graphs. These results are helpful in constructing stable networks with lower costs. The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.

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.