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

Jonas Hägglund, Klas Markström
2013 Journal of Combinatorics  
In particular, we prove that graphs 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.  ... 
doi:10.4310/joc.2013.v4.n4.a5 fatcat:kg57bn62xva7fck63hplfa7gk4

Shortest cycle covers and cycle double covers with large 2-regular subgraphs [article]

Jonas Hägglund, Klas Markstrøm
2013 arXiv   pre-print
In particular we prove that graphs 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.  ... 
arXiv:1306.3088v1 fatcat:d7awrx4x6feh7bogfbgk2u6le4

The traveling salesman problem on cubic and subcubic graphs [article]

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2011 arXiv   pre-print
In fact we prove constructively that for any cubic graph on n vertices a tour 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.  ... 
arXiv:1107.1052v1 fatcat:sbklbt5rebaytnvbpsd7atmd7q

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.  ... 

On the oriented perfect path double cover conjecture [article]

Behrooz Bagheri Gh., Behnaz Omoomi
2012 arXiv   pre-print
An oriented 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  ... 
arXiv:1207.1961v1 fatcat:3erwaz5ufvbbfg7g762zxgvv3u

Antipodal covers of strongly regular graphs

Aleksandar Jurišić
1998 Discrete Mathematics  
to affine planes 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  ... 
doi:10.1016/s0012-365x(97)00139-8 fatcat:shuni5ozqffhroi3psagupjgyy

On the number of dissimilar pfaffian orientations of graphs

Marcelo H. de Carvalho, Cláudio L. Lucchesi, U. S.R. Murty
2005 RAIRO - Theoretical Informatics and Applications  
An orientation D 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.  ... 
doi:10.1051/ita:2005005 fatcat:ulcsudujmbcxxltozazwbwiu7y

Classes of line graphs with small cycle double covers

Gary MacGillivray, Karen Seyffarth
2001 The Australasian Journal of Combinatorics  
The Small Cycle 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.  ... 
dblp:journals/ajc/MacGillivrayS01 fatcat:b24uo33rt5cdxjl355rcnazc64

The traveling salesman problem on cubic and subcubic graphs

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2012 Mathematical programming  
In fact we prove constructively that for any cubic graph on n vertices a tour 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.  ... 
doi:10.1007/s10107-012-0620-1 fatcat:34lq2cwdbrfqzntudb7v5t2q3u

Red-green-blue model

David B. Wilson
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 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  ... 
doi:10.1103/physreve.69.037105 pmid:15089445 fatcat:vvwe67j2d5cstlidkziq3cnihu

TSP on Cubic and Subcubic Graphs [chapter]

Sylvia Boyd, René Sitters, Suzanne van der Ster, Leen Stougie
2011 Lecture Notes in Computer Science  
The problem is 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.  ... 
doi:10.1007/978-3-642-20807-2_6 fatcat:donyaxftkfcmpd44bbthuyueay

On the perfect matching index of bridgeless cubic graphs [article]

Jean-Luc Fouquet
2009 arXiv   pre-print
,M_6 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.  ... 
arXiv:0904.1296v1 fatcat:ae4r4j4rdfd2xg3rfyklwetzam

An improved upper bound for the TSP in cubic 3-edge-connected graphs

David Gamarnik, Moshe Lewenstein, Maxim Sviridenko
2005 Operations Research Letters  
We consider the travelling salesman problem (TSP) problem on (the metric completion 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.  ... 
doi:10.1016/j.orl.2004.09.005 fatcat:fnzbstj7qbby5l6fjzyuzixeqy

On covers of graphs

Jana Maxová, Jaroslav Nešetřil
2004 Discrete Mathematics  
We consider two problems related to the cycle 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  ... 
doi:10.1016/s0012-365x(03)00314-5 fatcat:xe6wdho2gnhm3frfkfzpc7llgy

Treelike Snarks

Marién Abreu, Tomáš Kaiser, Domenico Labbate, Giuseppe Mazzuoccolo
2016 Electronic Journal of Combinatorics  
We study snarks whose edges cannot be 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.  ... 
doi:10.37236/6008 fatcat:zolqjvztd5gz3fpb6mz67xstay
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