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On line graphs and the hamiltonian index

1981
*
Discrete Mathematics
*

An extension of CI theorem of Chamhnd

doi:10.1016/0012-365x(81)90058-3
fatcat:evhgrfd7mzgulaytzz6zlk56mu
*and*Wall is obtained*and*, with it, a bound*on**the*lamiltoniau*index*h(G) of a connected*graph*G (other than a path) is determined. ... With every nonempty*graph*G there AS associated a*graph*L(G), called*the**line**graph*of G, ha+ng*the*property thar *&ere exists a*one*-to-*one*correspondence between E(G)*and*V(L(G)) such that two vertices ...*The**graph*(a) in Fig. 3 is such an example. We next note that Theorem 1 allows us to bound*the**hamiltonian**index*of any homogeneously traceable*graph*. @rm 3. ...##
###
A sharp lower bound on the signless Laplacian index of graphs with (κ,τ)-regular sets

2018
*
Special Matrices
*

A new lower bound

doi:10.1515/spma-2018-0007
fatcat:sj43sg4a65gyrj5zyopm57ewom
*on**the*largest eigenvalue of*the*signless Laplacian spectra for*graphs*with at least*one*(κ,τ)regular set is introduced*and*applied to*the*recognition of non-*Hamiltonian**graphs*or*graphs*...*The*paper also gives sufficient condition for a*graph*to be non*Hamiltonian*(or without a perfect matching). ... In this section we deduce lower bounds*on**the*signless Laplacian*index*of*the**line**graph*of a*graph*with a*Hamiltonian*cycle or a*graph*with a perfect matching. ...##
###
Hamiltonian iterated line graphs

2002
*
Discrete Mathematics
*

As an application, we show two methods for determining

doi:10.1016/s0012-365x(01)00442-3
fatcat:e52zopn7izbqrgg7hr4xl23ama
*the**hamiltonian**index*of a*graph**and*enhance various results*on**the**hamiltonian**index*known earlier. ... a simple consequence of Harary*and*Nash-Williams' result. ... Acknowledgements*The*authors thank*the*two anonymous referees who made invaluable comments*on*Theorem 18*and*who carefully checked previous versions of*the*manuscript. They also thank H.J. ...##
###
Asymptotically sharpening the s-Hamiltonian index bound
[article]

2021
*
arXiv
*
pre-print

For a non-negative integer s≤ |V(G)|-3, a

arXiv:2109.05660v1
fatcat:o3oet3fucjdpdg564vmm63mvyq
*graph*G is s-*Hamiltonian*if*the*removal of any k≤ s vertices results in a*Hamiltonian**graph*. ... Consequently, when s ≥ 5, this new upper bound for*the*s-*hamiltonian**index*implies that h_s(G) = o(ℓ(G)+s+1) as s →∞. ... Chartrand [9] showed that for every*graph*G ∈ G,*the**Hamiltonian**index*exists as a finite number,*and**the*characterization of*Hamiltonian**line**graphs*(Theorem 1.1) by Harary*and*Nash-Williams implies ...##
###
On n-Hamiltonian line graphs

1977
*
Journal of combinatorial theory. Series B (Print)
*

INTRODUCTIQN With each nonempty

doi:10.1016/0095-8956(77)90071-5
fatcat:qe45emzcwzaqlmog4w6c4qigwy
*graph*G*one*can associate a*graph*L (G) , called*the**line**graph*of G, with*the*property that there exists G*one*-to-*one*correspondence between*the*edge set E(G) of G*and*... Theorem 4 deals with*one*aspect of*the*relationship between iterated*line**graphs**and**the*property of being n-*Hamiltonian*, for arbitrary values of n > 1. ...##
###
Page 5752 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

Ryja¢ek [personal communication] has disproved

*the*conjecture:*The**line**graph*of a*graph*obtained from*the*Pe- tersen*graph*by adding at least*one*pendant edge to each vertex has connectivity three*and*... Summary: “It is proved that*the**Hamiltonian**index*of a connected*graph*other than a path is less than its diameter, which improves*the*results of P. A. Catlin et al. [J. ...##
###
Page 3051 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

Chen (1-ILCC-EL)
89f:05113 05C45 05C75
Lai, Hong-Jian (1-WYNS)

*On**the**Hamiltonian**index*. Discrete Math. 69 (1988), no. 1, 43-53. We denote*the**line**graph*of G by L(G). ... If either two edges or*one*edge*and**one*vertex are removed from G°,*the*resulting*graph*is*Hamiltonian*. G. Chartrand (1-WMI) ...##
###
Page 40 of Mathematical Reviews Vol. , Issue 95a
[page]

1995
*
Mathematical Reviews
*

Varma (“Super

*line**graphs*”, in Proceedings of*the*Seventh International Conference*on**Graph*Theory, Combinatorics, Algorithms*and*Appli- cations, to appear] defined*the*super*line**graph*. ... In this paper some properties of*the*super*line**graph*of*index*two of*the*hyper- cube Q, are established.*The*size of -44(Q,) is determined. ...##
###
Hamiltonian index is NP-complete

2011
*
Discrete Applied Mathematics
*

Finally, we show that some known upper

doi:10.1016/j.dam.2010.08.027
fatcat:qjtakjf3ivgenfx3xdodd6cki4
*and*lower bounds*on**the**hamiltonian**index*can be computed in polynomial time. ... Consequently,*the*corresponding problem to determine*the**hamiltonian**index*of a given*graph*is NP-hard. ... NP-completeness of*the**hamiltonian**index*Let G be a*graph*. ...##
###
On stability of the hamiltonian index under contractions and closures

2005
*
Journal of Graph Theory
*

*The*

*hamiltonian*

*index*of a

*graph*G is

*the*smallest integer k such that

*the*k-th iterated

*line*

*graph*of G is

*hamiltonian*. ... We r s t s h o w that, with

*one*exceptional case, adding an edge to a

*graph*cannot increase its

*hamiltonian*

*index*. ...

*The*

*hamiltonian*

*index*of a

*graph*G, denoted h(G), is

*the*smallest integer k such t h a t

*the*k-th iterated

*line*

*graph*L k (G) o f G is

*hamiltonian*. ...

##
###
On hamiltonian line-graphs

1968
*
Transactions of the American Mathematical Society
*

Let xtl

doi:10.1090/s0002-9947-1968-0231740-1
fatcat:psvp6zpwizcubn6hnubgqnvn5y
*and*xi2 be*lines*of £p-*(G) which join «¿ to*the*distinct endpoints of a*line*y¡ of Lp~i(Gl). We now claim that Lp~i(G) is a sequential*graph*so that LP~3(G) is*hamiltonian*. ... Now Fj,_4 has a cycle C containing all*the*points of//p_4*and*every*line*of /p_4 is incident with at least*one*point of C. ...*One*readily sees that*the**line*-*graph*of each of these two*graphs*is*hamiltonian*,*and**the*result is established for p=4. ...##
###
On the hamiltonian index of a graph

1993
*
Discrete Mathematics
*

Furthermore, a counterexample to a theorem of Chartrand

doi:10.1016/0012-365x(93)90313-i
fatcat:6zj2pxxvurgwhc7hvbs6gwsaxa
*and*Wall (1973) about*hamiltonian**index*of*graphs*with*hamiltonian*cyclic blocks was exhibited by Lai (1988) . ... It was claimed by Gould (1981) that if G is a connected*graph*of order at least 3 such that no bridge is incident to a vertex of degree 2*and*no path contains three or more consecutive vertices of degree ... For a connected*graph*G that is not a path we define*the**hamiltonian**index*of G (denoted by h(G)) as*the*smallest n such that L"(G) is*hamiltonian**graph*. ...##
###
On traceable iterated line graph and hamiltonian path index
[article]

2020
*
arXiv
*
pre-print

As applications, we use this characterization to give several upper bounds

arXiv:2012.14551v1
fatcat:grqh7hn7n5gujm4bkdz6uavvqa
*on**the**hamiltonian*path*index*of a*graph*. ... Liu,*Hamiltonian*iterated*line**graphs*, Discrete Math. 256 (2002) 407-422] gave a characterization of*the**graphs*G for which*the*n-th iterated*line**graph*L^n(G) is*hamiltonian*, for n≥2. ...*The*first*one*indicates a close relationship between EU k (L(G))*and*EU k+1 (G),*and**the*second*one*characterizes*graphs*with 2-iterated*line**graphs*are*hamiltonian*. ...##
###
An Optimal Algorithm for the k-Fixed-Endpoint Path Cover on Proper Interval Graphs

2009
*
Mathematics in Computer Science
*

In this paper we consider

doi:10.1007/s11786-009-0004-y
fatcat:g7nvnlf4hncdjmfcx22yssglp4
*the*k-fixed-endpoint path cover problem*on*proper interval*graphs*, which is a generalization of*the*path cover problem. ... This algorithm is based*on**the*Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval*graphs*. ... Thus, always*one*of*the**lines*7*and*10 is executed. ...##
###
Some problems related to hamiltonian line graphs
[chapter]

2007
*
Proceedings of the International Conference on Complex Geometry and Related Fields
*

Part of this paper summarizes some of

doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
*the*recent developments in*the*study of*hamiltonian**line**graphs**and**the*related*hamiltonian*claw-free*graphs*. ...*The*last section of this paper solves some problems*on**the**hamiltonian*like indices from a paper by Clark*and*Wormald in 1983. ...*The*purpose of this article is to summarize some of*the*recent development*on**the*study of*hamiltonian**line**graphs*,*hamiltonian*claw-free*graphs*,*and*to solve some of*the*problems from an earlier study ...
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