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Contraction theorems in Hamiltonian graph theory

1981
*
Discrete Mathematics
*

Also stated

doi:10.1016/0012-365x(81)90022-4
fatcat:mjztgawtdrddhao6nnnhjuabxa
*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*. ...##
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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

2008
*
Discrete Mathematics
*

Thomassen [Reflections on

doi:10.1016/j.disc.2007.06.009
fatcat:cqkdcmjxzrgatn7oio74jy5r6e
*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. ...##
###
The ubiquitous Petersen graph

1992
*
Discrete Mathematics
*

Wilson, The ubiquitous Petersen

doi:10.1016/0012-365x(92)90649-z
fatcat:j3ehmu27nndfbe5t5tj5gaku5m
*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*'. ...##
###
On characterization of Hamiltonian graphs

1978
*
Journal of combinatorial theory. Series B (Print)
*

A necessary and a sufficient condition are derived for a

doi:10.1016/s0095-8956(78)80009-4
fatcat:m5doj3ck3reqjl25ly673uhnru
*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. ...##
###
A note on edge-disjoint contractible Hamiltonian cycles in polyhedral maps

English

2014
*
Electronic Journal of Graph Theory and Applications
*

English

We present a necessary and sufficient condition for the existence of edge-disjoint

doi:10.5614/ejgta.2014.2.2.8
fatcat:iricxoyehfbpxguy3huiti4ojy
*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. ...##
###
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

2011
*
Advances in Applied Mathematics
*

The classical relationship between the Tutte polynomial of

doi:10.1016/j.aam.2011.02.004
fatcat:qvbj4cnjxnehdiwq4ogudltvpa
*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. ...##
###
Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs

1973
*
Journal of combinatorial theory. Series B (Print)
*

For all n ~> 3, regular n-valent nonHamiltonian non-n-edge-colorable

doi:10.1016/s0095-8956(73)80006-1
fatcat:czehutdzpnh35ltg5q6egzbthy
*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. ...##
###
A planar hypohamiltonian graph with 48 vertices

2007
*
Journal of Graph Theory
*

We present a planar hypohamiltonian

doi:10.1002/jgt.20241
fatcat:fxqt46ggkjgr7c6ngvccfg2zee
*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*. ...##
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On stability of the hamiltonian index under contractions and closures

2005
*
Journal of Graph Theory
*

We use this result to prove that neither the

doi:10.1002/jgt.20068
fatcat:v6p4xyhnzze4hlzhgzxsduikqy
*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 ...##
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Existence of dominating cycles and paths

1983
*
Discrete Mathematics
*

If a

doi:10.1016/0012-365x(83)90165-6
fatcat:uahgocys7jc4vbovpsjojkvppi
*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. ...
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