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An extension of CI theorem of Chamhnd 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. ...doi:10.1016/0012-365x(81)90058-3 fatcat:evhgrfd7mzgulaytzz6zlk56mu
A new lower bound 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. ...doi:10.1515/spma-2018-0007 fatcat:sj43sg4a65gyrj5zyopm57ewom
As an application, we show two methods for determining 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. ...doi:10.1016/s0012-365x(01)00442-3 fatcat:e52zopn7izbqrgg7hr4xl23ama
For a non-negative integer s≤ |V(G)|-3, a 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  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 ...arXiv:2109.05660v1 fatcat:o3oet3fucjdpdg564vmm63mvyq
INTRODUCTIQN With each nonempty 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. ...doi:10.1016/0095-8956(77)90071-5 fatcat:qe45emzcwzaqlmog4w6c4qigwy
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. ...
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) ...
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. ...
Finally, we show that some known upper 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. ...doi:10.1016/j.dam.2010.08.027 fatcat:qjtakjf3ivgenfx3xdodd6cki4
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. ...doi:10.1002/jgt.20068 fatcat:v6p4xyhnzze4hlzhgzxsduikqy
Let xtl 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. ...doi:10.1090/s0002-9947-1968-0231740-1 fatcat:psvp6zpwizcubn6hnubgqnvn5y
Furthermore, a counterexample to a theorem of Chartrand 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. ...doi:10.1016/0012-365x(93)90313-i fatcat:6zj2pxxvurgwhc7hvbs6gwsaxa
As applications, we use this characterization to give several upper bounds 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. ...arXiv:2012.14551v1 fatcat:grqh7hn7n5gujm4bkdz6uavvqa
In this paper we consider 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. ...doi:10.1007/s11786-009-0004-y fatcat:g7nvnlf4hncdjmfcx22yssglp4
Proceedings of the International Conference on Complex Geometry and Related Fields
Part of this paper summarizes some of 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 ...doi:10.1090/amsip/039/09 fatcat:pxde4ptk7jfptarou6gs2f5iqi
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