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Contraction theorems in Hamiltonian graph theory

C. Hoede, H.J. Veldman
1981 Discrete Mathematics  
Also stated in terms of contractibility are sufficient conditions for graphs to be traceable, Hamiltonian connected or 1-Hamiltonian, respectiveiy.  ...  We prove that a k-connected graph (k 2 2) is Hamiltonian if it is not contractible to one of a specified collection of graphs of order 2k + 1.  ...  We shall state a contraction theorem for I-Hamiltonian graphs.  ... 
doi:10.1016/0012-365x(81)90022-4 fatcat:mjztgawtdrddhao6nnnhjuabxa

Page 1850 of Mathematical Reviews Vol. , Issue 82e [page]

1982 Mathematical Reviews  
J. 82e:05095 Contraction theorems in Hamiltonian graph theory. Discrete Math. 34 (1981), no. 1, 61-67.  ...  For traceable and 1-Hamiltonian graphs the contraction theorems sharpen the corresponding analogues of (*), while equivalence is conjectured for Hamiltonian connected graphs.”  ... 

Page 3571 of Mathematical Reviews Vol. , Issue 91G [page]

1991 Mathematical Reviews  
In particular, Ore’s theorem is generalized to Hamiltonian cycles through any perfect matching.  ...  ISBN 0-444-88395-9 This welcome book presents the theory of Eulerian graphs and di- graphs in a form suitable for students and researchers in graph theory.  ... 

Page 4954 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
(1) and G can be contracted to Kz p_2 (p is odd) by contracting a complete m-partite graph Tin.n—p41 of order n—p+1 in G.  ...  Here an alternative version of this theorem is presented.” 94i:05060 05C45 05C40 Wei, Bing (D-TUB; Berlin) Hamiltonian paths and Hamiltonian connectivity in graphs.  ... 

Hamiltonian connected hourglass free line graphs

Li Dengxin, Hong-Jian Lai, Yehong Shao, Mingquan Zhan
2008 Discrete Mathematics  
Thomassen [Reflections on graph theory, J. Graph Theory 10 (1986) 309-324] conjectured that every 4-connected line graph is hamiltonian.  ...  Graph Theory 37 (2001) 125-136], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected.  ...  In this note, we shall improve Theorem 1.1 in the following form. Theorem 1.2. Every 3-connected, essentially 4-connected hourglass free line graph is hamiltonian connected.  ... 
doi:10.1016/j.disc.2007.06.009 fatcat:cqkdcmjxzrgatn7oio74jy5r6e

The ubiquitous Petersen graph

Gary Chartrand, Héctor Hevia, Robin J. Wilson
1992 Discrete Mathematics  
Wilson, The ubiquitous Petersen graph, Discrete Mathematics 100 (1992) 303-311.  ...  If G is a 3-connected 3-regular graph, then any set A of eleven vertices lies on at least three different cycles of G unless there is a contraction In [20] ErdGs mentioned Theorem 7 and attributed it  ...  It is hypohamiltonian, however, in that the removal of any vertex leaves a Hamiltonian graph. Thus, the Petersen graph is, in some sense, 'nearly Hamiltonian'.  ... 
doi:10.1016/0012-365x(92)90649-z fatcat:j3ehmu27nndfbe5t5tj5gaku5m

On characterization of Hamiltonian graphs

C. Hoede, H.J. Veldman
1978 Journal of combinatorial theory. Series B (Print)  
A necessary and a sufficient condition are derived for a graph to be non- Hamiltonian. 47 0095-8956]78]0251-0047502.00/0  ...  Cohen for pointing out shortcuts and shortcomings in the original version of this paper. Thanks are also due to A. v. d. Tuin for discussions on the paper.  ...  P2 Ql Q2 P2 Q1 @ % FIGURE 1 Theorem 7.2 in [1] states that every Hamiltonian graph is 2-connected and that every non-Hamiltonian 2-connected graph has a 0 subgraph.  ... 
doi:10.1016/s0095-8956(78)80009-4 fatcat:m5doj3ck3reqjl25ly673uhnru

A note on edge-disjoint contractible Hamiltonian cycles in polyhedral maps

Dipendu Maity, Ashish Kumar Upadhyay
2014 Electronic Journal of Graph Theory and Applications  
We present a necessary and sufficient condition for the existence of edge-disjoint contractible Hamiltonian cycles in the edge graph of polyhedral maps.  ...  Acknowledgement Authors are grateful to anonymous referee whose suggestions lead to substantial improvement in the presentation and content of the paper.  ...  In this note we present a necessary and sufficient condition for the existence of edge-disjoint contractible Hamiltonian cycles in the edge graph of a polyhedral map.  ... 
doi:10.5614/ejgta.2014.2.2.8 fatcat:iricxoyehfbpxguy3huiti4ojy

Page 5938 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews  
A contractible edge in a 3-connected graph is an edge whose contraction results in a 3-connected graph.  ...  {Reviewer’s remark: We note that references 7 and 8 are inter- changed in the text. Also, reference 8, “A Dirac-type theorem on squares of graphs” by the author, has appeared [J. Graph Theory  ... 

Page 2757 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
(l-MEMP; Memphis, TN) Contractibility techniques as a closure concept. (English summary) J. Graph Theory 43 (2003), no. 1, 37-48. In an earlier paper, the first author [J. Combin. Theory Ser.  ...  Wojcicka conjectured that every 3-domination-critical graph with 6 > 2 has a Hamiltonian cycle [E. Wojcicka, J. Graph Theory 14 (1990), no. 2, 205-215; MR 91c:05120].  ... 

The Tutte–Potts connection in the presence of an external magnetic field

Joanna A. Ellis-Monaghan, Iain Moffatt
2011 Advances in Applied Mathematics  
The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines.  ...  Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications.  ...  into this theory in some way.  ... 
doi:10.1016/j.aam.2011.02.004 fatcat:qvbj4cnjxnehdiwq4ogudltvpa

Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs

G.H.J. Meredith
1973 Journal of combinatorial theory. Series B (Print)  
For all n ~> 3, regular n-valent nonHamiltonian non-n-edge-colorable graphs with an even ntunber of vertices are constructed. For n @ 5, 6, or 7, these graphs are n-connected.  ...  THEOREM Suppose G has a Hamiltonian cycle C. C must contain 2(n --1) a~bjtype edges, and therefore 2bic~-type edges. Thus contracting K to k contracts C to a Hamiltonian cycle of H.  ...  G, is the (necessarily unique) graph from which H,~ is obtained by ten simultaneous contractions as in Theorem 3. H, is nonHamiltonian, so by Theorem 1 G~ is nonHamiltonian.  ... 
doi:10.1016/s0095-8956(73)80006-1 fatcat:czehutdzpnh35ltg5q6egzbthy

A planar hypohamiltonian graph with 48 vertices

Carol T. Zamfirescu, Tudor I. Zamfirescu
2007 Journal of Graph Theory  
We present a planar hypohamiltonian graph on 48 vertices, and derive some consequences.  ...  Journal of Graph Theory DOI 10.1002/jgt  ...  Thus, if in a graph embedded in the plane every face but one is an i-gon with i = 2 modulo 3, then the graph is not hamiltonian.  ... 
doi:10.1002/jgt.20241 fatcat:fxqt46ggkjgr7c6ngvccfg2zee

On stability of the hamiltonian index under contractions and closures

Liming Xiong, Zden?k Ryj�?ek, Hajo Broersma
2005 Journal of Graph Theory  
We use this result to prove that neither the contraction of an A G (F)-contractible subgraph F of a graph G nor the closure operation performed on G (if G is claw-free) a ects the value of the hamiltonian  ...  The hamiltonian index of a graph G is the smallest integer k such that the k-th iterated line graph of G is hamiltonian.  ...  Thus, Theorem 7 implies that the hamiltonian index is stable under contraction of a collapsible subgraph. 4 The hamiltonian index of a claw-free graph is stable under the closure In this section we assume  ... 
doi:10.1002/jgt.20068 fatcat:v6p4xyhnzze4hlzhgzxsduikqy

Existence of dominating cycles and paths

H.J. Veldman
1983 Discrete Mathematics  
If a graph G contains a D-cycle (D-path), the n its edge graph t(G) has a hamiltonian cycle (hamiltonian path).  ...  Necessary conditions and sufficient conditions are obtained for graphs to have a D-cycle or D-path. They are analogous to known conditions for the existence of hamiltonian cycles or paths.  ...  F Giibel for his help in the preparation of this paper.  ... 
doi:10.1016/0012-365x(83)90165-6 fatcat:uahgocys7jc4vbovpsjojkvppi
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