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Pancyclicity of hamiltonian line graphs

1995
*
Discrete Mathematics
*

Let G be a

doi:10.1016/0012-365x(94)00220-d
fatcat:6dwbhecclbdj3esounacxksxda
*graph**of*order n with 3(G)~>600 n 1/a such that L(G) is*hamiltonian*. Then L(G) is*pancyclic*. ... If C is a circuit*of*G, then I(C) denotes the set*of*edges*of*G incident with at least one vertex*of*C. We write t(C) for II(C)l. Harary and Nash-Williams [8] characterized*hamiltonian**line**graphs*. ... A lower bound for f(n) We construct a family*of**graphs*with*hamiltonian*but not*pancyclic**line**graphs*in order to obtain a lower bound for f(n). ...##
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A characterization of pancyclic complements of line graphs

2005
*
Discrete Mathematics
*

We characterize

doi:10.1016/j.disc.2004.02.024
fatcat:vat2lmpmzred5e3oijlzk5q5li
*graphs*G such that the complements*of*their*line**graphs*are*pancyclic*. ... We characterize all*graphs*such that the complement*graphs**of*their*line**graphs*are*pancyclic*or bi-*pancyclic*. ... For the class*of*complements*of**line**graphs*, our characterization provides the following linear time algorithm recognizing*pancyclic*complements*of**line**graphs*. Theorem 2.4. ...##
###
Page 2783 of Mathematical Reviews Vol. , Issue 85g
[page]

1985
*
Mathematical Reviews
*

*graphs*, the

*line*

*graph*and the total

*graph*

*of*a

*graph*. ... For some

*of*these operations we find that the spectrum

*of*the

*graph*so obtained is always infinite (direct sum,

*line*and total

*graph*). ...

##
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Vertex-pancyclicity of edge-face-total graphs

2004
*
Discrete Applied Mathematics
*

; (2) the

doi:10.1016/j.dam.2003.07.008
fatcat:fxlo3z7z3japzd4ji6qyij4zme
*line**graph**of*a 2-connected plane*graph*with at most one face*of*degree ¿ 4 is*pancyclic*. ... In this paper we prove that (1) the edge-face-total*graph**of*a 2-connected plane*graph*is vertex-*pancyclic*and there exists a connected plane*graph*G with cut vertices such that r(G) is non-*Hamiltonian*... However, it is unknown if the*line**graph**of*every 3-connected planar*graph*is*Hamiltonian*. ...##
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Some problems related to hamiltonian line graphs
[chapter]

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

Part

doi:10.1090/amsip/039/09
fatcat:pxde4ptk7jfptarou6gs2f5iqi
*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 ...##
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Pancyclic zero divisor graph over the ring Z_n[i]
[article]

2018
*
arXiv
*
pre-print

Let Γ(Z_n[i]) be the zero divisor

arXiv:1806.08261v1
fatcat:hwn6mwf5b5fs7dpr7ni3ntorsi
*graph*over the ring Z_n[i]. In this article, we study*pancyclic*properties*of*Γ(Z_n[i]) and Γ(Z_n[i]) for different n. ... Also, we prove some results in which L(Γ(Z_n[i])) and L(Γ(Z_n[i])) to be*pancyclic*for different values*of*n. ... Suppose p " 2 and q ą 3, then ΓpZ 2q q is a star*graph*and its*line**graph*LpΓpZ 2q qq is a complete*graph**of*order q´1. So LpΓpZ 2q qq is*pancyclic*. ...##
###
Edge degree conditions for subpancyclicity in line graphs

1998
*
Discrete Mathematics
*

In [5] the following conjecture is made: If G is a

doi:10.1016/s0012-365x(97)00257-4
fatcat:f22yj76fafazdndu6dryu4oxze
*graph*such that the degree sum*of*any pair*of*adjacent vertices in G is greater than ( 8v/~ 1 + 1)/2, then the*line**graph*L(G)*of*G is*pancyclic*whenever ... L(G) is*Hamiltonian*, unless G is isomorphic to C4, C5, or the Petersen*graph*. ... Also, thanks the referee for his careful correction*of*his manuscript. This work was supported in part by the project (Mathematical Methods*of*Technology)*of*the Flemish government. ...##
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Alternating-pancyclism in 2-edge-colored graphs

2020
*
Discussiones Mathematicae Graph Theory
*

. , G k be a collection

doi:10.7151/dmgt.2350
fatcat:2g5zzu536fbcrlsw6saz7excpm
*of*pairwise vertex disjoint 2-edge-colored*graphs*. ... The colored generalized sum*of*G 1 , . . . , G k , denoted by ⊕ k i=1 G i , is the set*of*all 2-edge-colored*graphs*G such that: (i) V (G) = ... Acknowledgements We would like to thank the anonymous referees for many comments which helped to improve the presentation*of*this paper. ...##
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s-Vertex Pancyclic Index

2011
*
Graphs and Combinatorics
*

If the removal

doi:10.1007/s00373-011-1052-z
fatcat:4nqlu3xlnvggtdxr2pnkvv7iuq
*of*at most s vertices in G results in a vertex*pancyclic**graph*, we say G is an s-vertex*pancyclic**graph*. Let G be a simple connected*graph*that is not a path, cycle or K 1,3 . ... The s-vertex*pancyclic*index*of*G, written vp s (G), is the least nonnegative integer m such that L m (G) is s-vertex*pancyclic*. We show Z. ... A*graph*is s-*pancyclic*if the removal*of*at most s vertices results in a*pancyclic**graph*; a*graph*is s-*hamiltonian*if the removal*of*at most s vertices results in a*hamiltonian**graph*. ...##
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Pancyclicity of claw-free hamiltonian graphs

1999
*
Discrete Mathematics
*

It follows directly that under the same condition a

doi:10.1016/s0012-365x(99)90147-4
fatcat:i46ogigd6zev5jynkotl5ahaiq
*hamiltonian*claw-free*graph*is*pancyclic*. ... A*graph*G on n vertices is called subpano'clic if it contains cycles*of*' every length k with 3 <~k ~c(G), where c(G) denotes the length*of*a longest cycle in G; if moreover c(G)= n, then G is called*pancyclic*... Before quoting a result from [12] we define, for a*graph*G with at least one edge, The*graph*Gp is a*hamiltonian**line**graph*. ...##
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Pancyclicity of claw-free hamiltonian graphs

1999
*
Discrete Mathematics
*

It follows directly that under the same condition a

doi:10.1016/s0012-365x(98)00278-7
fatcat:dmaterkf7vacrkvwvox6mahkme
*hamiltonian*claw-free*graph*is*pancyclic*. ... A*graph*G on n vertices is called subpano'clic if it contains cycles*of*' every length k with 3 <~k ~c(G), where c(G) denotes the length*of*a longest cycle in G; if moreover c(G)= n, then G is called*pancyclic*... Before quoting a result from [12] we define, for a*graph*G with at least one edge, The*graph*Gp is a*hamiltonian**line**graph*. ...##
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Frequency partitions: forcibly pancyclic and forcibly nonhamiltonian degree sequences

1977
*
Discrete Mathematics
*

A graphic sequence

doi:10.1016/0012-365x(77)90049-8
fatcat:preoi36g65flpksw5hs2mephzy
*of*length p is sa\id lo be /orci&*pancyclic*if Gvery realization*of*it has cycles af all lengths n, 3 g n 6 p. ... ., n,) is the frequency partition*of*the*graph*CT and as well as l\f rr [3]. We consider frequency partiiions p = nl + nr + * * * + n,. with n ,> nr 3 --* 23 tr, 3 I. ... If G is*hamiltonian*then every*hamiltonian*cycle has to contain the unique*line*e f: b,b, say*of*CS). Now, the number*of*vertices*of*B other than b,, b2 is 3r --1 and /A / = 3t. ...##
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Eulerian and Hamiltonian Graphs
[chapter]

2000
*
Universitext
*

*of*

*graphs*, now called Eulerian

*graphs*and

*Hamiltonian*

*graphs*. ... In fact, the two early discoveries which led to the existence

*of*

*graphs*arose from puzzles, namely, the Konigsberg Bridge Problem and

*Hamiltonian*Game, and these puzzles also resulted in the special types ... This is a contradiction and hence G is

*Hamiltonian*.

*Pancyclic*

*Graphs*Definition: A

*graph*G

*of*order n(≥ 3) is

*pancyclic*if G contains all cycles

*of*lengths from 3 to n. ...

##
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Cycle Lengths in Hamiltonian Graphs with a Pair of Vertices Having Large Degree Sum

2010
*
Graphs and Combinatorics
*

A

doi:10.1007/s00373-010-0915-z
fatcat:32755gehcvd73ldpfghacgwh5y
*graph**of*order n is said to be*pancyclic*if it contains cycles*of*all lengths from three to n. ... Let G be a*hamiltonian**graph*and let x and y be vertices*of*G that are consecutive on some*hamiltonian*cycle in G. ... Along these*lines*, it may be*of*interest to investigate the effect*of*larger sets*of*vertices with high degree sum on the cycle spectrum*of*a*hamiltonian**graph*. ...##
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Cycle Extendability of Hamiltonian Interval Graphs

2006
*
SIAM Journal on Discrete Mathematics
*

A

doi:10.1137/s0895480104441450
fatcat:dgaxqlvokbgyzcejk7i5nb7rf4
*graph*G*of*order n is*pancyclic*if it contains cycles*of*all lengths from 3 to n. ... This supports a conjecture*of*Hendry that all*Hamiltonian*chordal*graphs*are cycle extendable. ... For example, a*graph*G*of*order n is*pancyclic*if it contains cycles*of*all lengths from 3 to n. Clearly, every*pancyclic**graph*is*Hamiltonian*, but the converse is not true. ...
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