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Shortest cycle covers and cycle double covers with large 2-regular subgraphs

2013
*
Journal of Combinatorics
*

In particular, we prove that graphs

doi:10.4310/joc.2013.v4.n4.a5
fatcat:kg57bn62xva7fck63hplfa7gk4
*with**perfect*matching index at most 4 have cycle*covers**of**length*4 3 m and satisfy the (1, 2)*covering*conjecture*of*Zhang, and that graphs*with*large circumference ... have cycle*covers**of**length*close to 4 3 m. ... If e ∈ F then e is*covered*by exactly one*perfect*matching in M and if e ∈ E(G) \ F it is*covered*by exactly two elements*of*M. Hence M is a*perfect*matching*cover**of*G*with**four**perfect*matchings. ...##
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Shortest cycle covers and cycle double covers with large 2-regular subgraphs
[article]

2013
*
arXiv
*
pre-print

In particular we prove that graphs

arXiv:1306.3088v1
fatcat:d7awrx4x6feh7bogfbgk2u6le4
*with**perfect*matching index at most 4 have cycle*covers**of**length*4/3m and satisfy the (1,2)-*covering*conjecture*of*Zhang, and that graphs*with*large circumference have ... cycle*covers**of**length*4/3m+o(m). ... If e ∈ F then e is*covered*by exactly one*perfect*matching in M and if e ∈ E(G) \ F it is*covered*by exactly two elements*of*M. Hence M is a*perfect*matching*cover**of*G*with**four**perfect*matchings. ...##
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The traveling salesman problem on cubic and subcubic graphs
[article]

2011
*
arXiv
*
pre-print

In fact we prove constructively that for any cubic graph on n vertices a tour

arXiv:1107.1052v1
fatcat:sbklbt5rebaytnvbpsd7atmd7q
*of**length*4n/3-2 exists, which also implies the 4/3 conjecture, as an upper bound, for this class*of*graph-TSP. ... We present the first algorithm for cubic graphs*with*approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is*of*independent interest. ... If it does not contain edge st, then the Eulerian subgraph either contains exactly one copy*of*each*of*the*four*edges incident to st or three*of*these*four*edges*doubled*. ...##
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Page 7225 of Mathematical Reviews Vol. , Issue 96m
[page]

1996
*
Mathematical Reviews
*

[Wallis, Walter D.] (1-SIL; Carbondale, IL); Yu, Qinglin

*Perfect**double**covers**with**paths**of**length**four*. (English summary) J. Graph Theory 21 (1996), no. 2, 187-197. ... Moreover, if every*path*ina PPDC F is*of**length*k, F is called a k-regular*perfect**path**double**cover*(k-RPPDC). Li proved that every simple graph possesses a PPDC. ...##
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On the oriented perfect path double cover conjecture
[article]

2012
*
arXiv
*
pre-print

An oriented

arXiv:1207.1961v1
fatcat:3erwaz5ufvbbfg7g762zxgvv3u
*perfect**path**double**cover*( OPPDC)*of*a graph G is a collection*of*directed*paths*in the symmetric orientation G_s*of*G such that each edge*of*G_s lies in exactly one*of*the*paths*and each ... to this conjecture is at least*four*. ... A*perfect**path**double**cover*(PPDC)*of*a graph G is a collection P*of**paths*in G such that each edge*of*G belongs to exactly two members*of*P and each vertex*of*G occurs exactly twice as an end*of*a*path*...##
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Antipodal covers of strongly regular graphs

1998
*
Discrete Mathematics
*

to affine planes

doi:10.1016/s0012-365x(97)00139-8
fatcat:shuni5ozqffhroi3psagupjgyy
*with*a parallel class deleted. ... Antipodal*covers**of*strongly regular graphs which are not necessarily distance-regular are studied. The structure*of*short cycles in an antipodal*cover*is considered. ... The complete bipartite graph Km, m*with*a*perfect*matching deleted is a triangle free distance-regular antipodal*double*-*cover**of*Kin, so, by Corollary 3.2, its line graph is an antipodal*double*-*cover**of*...##
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On the number of dissimilar pfaffian orientations of graphs

2005
*
RAIRO - Theoretical Informatics and Applications
*

An orientation D

doi:10.1051/ita:2005005
fatcat:ulcsudujmbcxxltozazwbwiu7y
*of*G is Pfaffian if, for every conformal even circuit C, the number*of*edges*of*C whose directions in D agree*with*any prescribed sense*of*orientation*of*C is odd. ... Annals*of*Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.] A matching*covered*graph is a nontrivial connected graph in which every edge is in some*perfect*matching. ... A removable ear*of*G is either a single or a*double*ear which is removable. A removable*double*ear in which both constituent*paths*have*length*one is a removable doubleton. ...##
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Classes of line graphs with small cycle double covers

2001
*
The Australasian Journal of Combinatorics
*

The Small Cycle

dblp:journals/ajc/MacGillivrayS01
fatcat:b24uo33rt5cdxjl355rcnazc64
*Double**Cover*Conjectnre, due to J .A. Bondy, states that every simple bridgeless graph on n vertices has a cycle*double**cover**with*at most (n -1) cycles. ... In this article, we prove that the conjecture holds for line graphs*of*a number*of*types*of*graphs; specifically line graphs*of*complete graphs, line graphs*of*complete bipartite graphs, and line graphs ... r:271-2Y2j' Then C 2 is a CDC*with*3(q -1) cycles*of**length**four*. ...##
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The traveling salesman problem on cubic and subcubic graphs

2012
*
Mathematical programming
*

In fact we prove constructively that for any cubic graph on n vertices a tour

doi:10.1007/s10107-012-0620-1
fatcat:34lq2cwdbrfqzntudb7v5t2q3u
*of**length*4n/3 − 2 exists, which also implies the 4/3-conjecture, as an upper bound, for this class*of*graph-TSP. ... We present the first algorithm for cubic graphs*with*approximation ratio 4/3. The proof uses polyhedral techniques in a surprising way, which is*of*independent interest. ... If it does not contain edge st, then the Eulerian subgraph either contains exactly one copy*of*each*of*the*four*edges incident to st or three*of*these*four*edges*doubled*. ...##
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Red-green-blue model

2004
*
Physical Review E
*

We find that when the boundary conditions are "flat", the red-green-blue loops are closely related to SLE_4 and

doi:10.1103/physreve.69.037105
pmid:15089445
fatcat:vvwe67j2d5cstlidkziq3cnihu
*double*-dimer loops, which are the loops formed by superimposing two dimer*coverings**of*the ... We experimentally study the red-green-blue model, which is a sytem*of*loops obtained by superimposing three dimer*coverings*on offset hexagonal lattices. ... An RGB configuration is a system*of*the loops on a region*of*the triangular lattice, which is obtained by superimposing three*perfect*matchings (or dimer*coverings*) on offset hexagonal lattices as shown ...##
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TSP on Cubic and Subcubic Graphs
[chapter]

2011
*
Lecture Notes in Computer Science
*

The problem is

doi:10.1007/978-3-642-20807-2_6
fatcat:donyaxftkfcmpd44bbthuyueay
*of*interest because*of*its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values*of*the TSP and ... We study the Travelling Salesman Problem (TSP) on the metric completion*of*cubic and subcubic graphs, which is known to be NP-hard. ... subgraphs which, together*with*the*double*spanning tree, result in k GSTP tours*with*average*length*at most 4/3n. ...##
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On the perfect matching index of bridgeless cubic graphs
[article]

2009
*
arXiv
*
pre-print

,M_6

arXiv:0904.1296v1
fatcat:ae4r4j4rdfd2xg3rfyklwetzam
*of*G*with*the property that every edge*of*G is contained in exactly two*of*them and Berge conjectured that its edge set can be*covered*by 5*perfect*matchings. ... The set*of*graphs*with**perfect*matching index 4 seems interesting and we give some informations on this class. ... If Fulkerson's conjecture were true, then deleting one*of*the*perfect*matchings from the*double**cover*would result in a*covering**of*the graph by 5*perfect*matchings. ...##
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An improved upper bound for the TSP in cubic 3-edge-connected graphs

2005
*
Operations Research Letters
*

We consider the travelling salesman problem (TSP) problem on (the metric completion

doi:10.1016/j.orl.2004.09.005
fatcat:fnzbstj7qbby5l6fjzyuzixeqy
*of*) 3-edge-connected cubic graphs. ... These graphs are interesting because*of*the connection between their optimal solutions and the subtour elimination LP relaxation. ... Acknowledgements We thank the anonymous referees*of*this paper for providing comments that improved the exposition*of*this paper. ...##
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On covers of graphs

2004
*
Discrete Mathematics
*

We consider two problems related to the cycle

doi:10.1016/s0012-365x(03)00314-5
fatcat:xe6wdho2gnhm3frfkfzpc7llgy
*double**cover*(CDC) conjecture for graphs: the oriented*perfect**path**double**cover*(OPPDC) and the oriented faithful cycle*cover*. ... Acknowledgements This research is supported by the project LN00A056*of*the Czech Ministry*of*Education and by FRV Ä S 1918. We thank Adrian Bondy who informed us about the Ref. [5] . ... Oriented*perfect**path**double**cover*An oriented*perfect**path**double**cover*(OPPDC)*of*a graph G is a collection*of*oriented*paths*in the symmetric orientation S(G) such that each edge*of*S(G) lies in exactly ...##
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Treelike Snarks

2016
*
Electronic Journal of Combinatorics
*

We study snarks whose edges cannot be

doi:10.37236/6008
fatcat:zolqjvztd5gz3fpb6mz67xstay
*covered*by fewer than five*perfect*matchings. Esperet and Mazzuoccolo found an infinite family*of*such snarks, generalising an example provided by Hägglund. ... In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number $\phi_C (G)\ge5$ and admit a 5-cycle*double**cover*. ... [14] ) is defined as a bridgeless cubic graph*with*edge chromatic number equal to*four*that contains no circuits*of**length*at most*four*and no non-trivial 3-edge cuts. ...
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