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### Pancyclicity of hamiltonian line graphs

E. van Blanken, J. van den Heuvel, H.J. Veldman
1995 Discrete Mathematics
Let G be a 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).  ...

### A characterization of pancyclic complements of line graphs

Jiping Liu
2005 Discrete Mathematics
We characterize 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).  ...

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

Wei-Fan Wang
2004 Discrete Applied Mathematics
; (2) the 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.  ...

### 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 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  ...

### Pancyclic zero divisor graph over the ring Z_n[i] [article]

Ravindra Kumar, Om Prakash
2018 arXiv   pre-print
Let Γ(Z_n[i]) be the zero divisor 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

Liming Xiong
1998 Discrete Mathematics
In [5] the following conjecture is made: If G is a 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.  ...

### Alternating-pancyclism in 2-edge-colored graphs

Narda Cordero-Michel, Hortensia Galeana-Sánchez
2020 Discussiones Mathematicae Graph Theory
. , G k be a collection 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.  ...

### s-Vertex Pancyclic Index

Zhang Lili, Yehong Shao, Guihai Chen, Xinping Xu, Ju Zhou
2011 Graphs and Combinatorics
If the removal 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.  ...

### Pancyclicity of claw-free hamiltonian graphs

H. Trommel, H.J. Veldman, A. Verschut
1999 Discrete Mathematics
It follows directly that under the same condition a 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.  ...

### Pancyclicity of claw-free hamiltonian graphs

H Trommel
1999 Discrete Mathematics
It follows directly that under the same condition a 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.  ...

### Frequency partitions: forcibly pancyclic and forcibly nonhamiltonian degree sequences

Vasanti N. Bhat-Nayak, Ranjan N. Naik, S.B. Rao
1977 Discrete Mathematics
A graphic sequence 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.  ...

### Eulerian and Hamiltonian Graphs [chapter]

R. Balakrishnan, K. Ranganathan
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.  ...

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

Michael Ferrara, Michael S. Jacobson, Angela Harris
2010 Graphs and Combinatorics
A 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.  ...

### Cycle Extendability of Hamiltonian Interval Graphs

Guantao Chen, Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson
2006 SIAM Journal on Discrete Mathematics
A 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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