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A note on shortest circuit cover of 3-edge colorable cubic signed graphs [article]

Ronggui Xu, Jiaao Li, Xinmin Hou
<span title="2022-04-12">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A sign-circuit coverof a signed graph (G, σ) is a family of sign-circuits which covers all edges of (G, σ).  ...  In this paper, we show that every flow-admissible 3-edge colorable cubic signed graph (G, σ) has a sign-circuit cover with length at most 20/9 |E(G)|.  ...  The problem to determine the optimal upper bound for the shortest sign-circuit cover of 3-edge colorable cubic signed graph remains open.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2204.05865v1">arXiv:2204.05865v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ua3e5rzpl5dbrlzd4xqimvhmye">fatcat:ua3e5rzpl5dbrlzd4xqimvhmye</a> </span>
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Circuit covers of cubic signed graphs

Yezhou Wu, Dong Ye
<span title="2018-02-01">2018</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
A circuit cover with the smallest total length is called a shortest circuit cover of (G,σ) and its length is denoted by scc(G,σ).  ...  The definition of a circuit of signed graph comes from the signed-graphic matroid. A circuit cover of (G,σ) is a family of circuits covering all edges of (G,σ).  ...  Shortest circuit covers In this section, we consider the shortest circuit covers of cubic signed graphs. Let , σ) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.22238">doi:10.1002/jgt.22238</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/da2s7rfr4nhz3b5gwimb6c5oi4">fatcat:da2s7rfr4nhz3b5gwimb6c5oi4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200911235247/https://arxiv.org/pdf/1609.03620v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3e/64/3e64a45a360b6ca0cba9ec414754c1b4e213ef9a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.22238"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

All-shortest-path 2-interval routing is NP-complete

Kai Wang, Rui Wang, Yanyan Liu
<span title="2009-03-26">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5oxn7l5nevejhflmckad42qrri" style="color: black;">Journal of Applied Mathematics and Computing</a> </i> &nbsp;
In this paper, we close the open case of k = 2 by showing that it is NP-complete to decide whether a graph admits an all-shortest-path 2-IRS.  ...  All of the problems related to single-shortest-path k-IRS have already been shown to be NP-complete.  ...  Discussion We have proved that to recognize networks that admit all-shortest-path 2-IRS (2-SIRS) is NP-complete for unweighted graphs, and therefore also for weighted graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s12190-009-0265-2">doi:10.1007/s12190-009-0265-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6pxejit4uvhsxo3dnnjqsehwby">fatcat:6pxejit4uvhsxo3dnnjqsehwby</a> </span>
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1-Factor and Cycle Covers of Cubic Graphs

Eckhard Steffen
<span title="2014-04-14">2014</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
Furthermore, if μ_3(G) = 0, then 2 μ_3(G) is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs.  ...  Cubic graphs with μ_4(G) = 0 have a 4-cycle cover of length 4/3 |E(G)| and a 5-cycle double cover. These graphs also satisfy two conjectures of Zhang.  ...  We prove some new upper bounds for the length of a shortest cycle covers of bridgeless cubic graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.21798">doi:10.1002/jgt.21798</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kxcbm7uj2bei7an2dimg3yaafm">fatcat:kxcbm7uj2bei7an2dimg3yaafm</a> </span>
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Circuit Double Covers of Graphs [chapter]

Cun-Quan Zhang
<span title="">2016</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vpgq5ivbqrhrjimehi5p4yyvum" style="color: black;">Problem Books in Mathematics</a> </i> &nbsp;
Contents Appendix B Snarks, Petersen graph 252 B.1 3-edge-coloring of cubic graphs, snarks 252 B.1.1 Parity lemma 253 B.1.2 Snarks 253 B.1.3 Construction of snarks 254 B.1.4 Girths and bonds of snarks  ...  162 14 Shortest cycle covers 163 14.1 Shortest cover and double cover 163 14.2 Minimum eulerian weight 166 14.3 3-even subgraph covers 168 14.3.1 Basis of cycle space 168 14.3.2 3-even subgraph  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-31940-7_16">doi:10.1007/978-3-319-31940-7_16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/peig3f6abrcfpe2edgdq5fitje">fatcat:peig3f6abrcfpe2edgdq5fitje</a> </span>
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Circuit extension and circuit double cover of graphs

Zhengke Miao, Dong Ye, Cun-Quan Zhang
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let G be a cubic graph and C be a circuit. An extension of C is a circuit D such that V (C) ⊆ V (D) and E(C ) = E(D).  ...  It was proved that if every circuit is extendable for every bridgeless cubic graph, then the circuit double cover conjecture is true (Kahn, Robertson, Seymour 1987) .  ...  Let G be a bridgeless cubic graph and C be a circuit of G. Then G has a circuit double cover which contains C .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2013.06.019">doi:10.1016/j.disc.2013.06.019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sky6nz3zpzaoxo6deuh6zuv22e">fatcat:sky6nz3zpzaoxo6deuh6zuv22e</a> </span>
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Cycle cover property and CPP=SCC property are not equivalent

Romeo Rizzi
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The shortest cycle cover problem (SCC) asks for a family C of circuits of G such that each edge is in some circuit of C and the total length of all circuits in C is as small as possible.  ...  Graph G is said to have the cycle cover property if for every Eulerian 1; 2-weighting w : E(G) → {1; 2} there exists a family C of circuits of G such that every edge e is in precisely we circuits of C.  ...  Let hk and xy be any two edges of G. Graph G is called an hk; xy-counterexample if there exists a shortest cycle cover C of G with hk; xy ∈ F G ( C) and a bad w G ∈ W G with w(hk) = w(xy) = 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(02)00590-3">doi:10.1016/s0012-365x(02)00590-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/23vgreqjx5ez5ph5qnqkyujhii">fatcat:23vgreqjx5ez5ph5qnqkyujhii</a> </span>
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Shortest coverings of graphs with cycles

Jean Claude Bermond, Bill Jackson, François Jaeger
<span title="">1983</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
ACKNOWLEDGMENTS We wish to thank Brian Alspach, Simon Fraser University, and the Canadian N.R.C. for support during the Workshop on Cycles (1982) at which much of this work was carried out.  ...  Clearly SY is a vertex cycle cover of G and I(g) < y (I V(G)/ -1). COVERING OF THE VERTICES OF A STRONG DIGRAPH WITH CIRCUITS In this section, circuit means "directed circuit."  ...  The length of a shortest cycle cover of G is equal to ! 1 E(G)1 .ProoJ As already seen in Subsection 1.3, the length of a shortest cycle cover of G is at least 4 1 E(G)1 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0095-8956(83)90056-4">doi:10.1016/0095-8956(83)90056-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/c6kpdomrzng5tjz3sh36qlukgi">fatcat:c6kpdomrzng5tjz3sh36qlukgi</a> </span>
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Short cycle covers on cubic graphs using chosen 2-factor [article]

Barbora Candráková, Robert Lukoťka
<span title="2015-09-24">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that every bridgeless cubic graph G with m edges has a cycle cover of length at most 1.6 m.  ...  Moreover, if G does not contain any intersecting circuits of length 5, then G has a cycle cover of length 212/135 · m ≈ 1.570 m and if G contains no 5-circuits, then it has a cycle cover of length at most  ...  Every bridgeless graph with m edges has cycle cover of length at most 1.4m. The shortest cycle cover of the Petersen graph has length 21 and consists of 4 cycles.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1509.07430v1">arXiv:1509.07430v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pj3gm647avevvoeag4z7zc7huy">fatcat:pj3gm647avevvoeag4z7zc7huy</a> </span>
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On semiextensions and circuit double covers

Enrique García Moreno Esteva, Tommy R. Jensen
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
It is shown that our conjecture implies a strong form of the circuit double cover conjecture.  ...  We prove that the conjecture is equivalent to its restriction to cubic graphs, and we show that it holds for every cycle which is a spanning subgraph of the given graph.  ...  Acknowledgments We are very grateful to those people who offered valuable comments on earlier drafts of the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jctb.2006.08.002">doi:10.1016/j.jctb.2006.08.002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ii6nzbu3lfhdli7bfdmytjuhbe">fatcat:ii6nzbu3lfhdli7bfdmytjuhbe</a> </span>
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Girth, oddness, and colouring defect of snarks [article]

Ján Karabáš and Edita Máčajová and Roman Nedela and Martin Škoviera
<span title="2022-03-16">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our aim is to examine the relationship of colouring defect to oddness, an extensively studied measure of uncolourability of cubic graphs, defined as the smallest number of odd circuits in a 2-factor.  ...  The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings.  ...  APVV-19-0308 of Slovak Research and Development Agency. The first and the third author were partially supported by the grant VEGA 2/0078/20 of Slovak Ministry of Education.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.12205v3">arXiv:2106.12205v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yod7bzzwafhp7gs4ykn34446xu">fatcat:yod7bzzwafhp7gs4ykn34446xu</a> </span>
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A note on Berge–Fulkerson coloring

Rongxia Hao, Jianbing Niu, Xiaofeng Wang, Cun-Quan Zhang, Taoye Zhang
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
The Berge-Fulkerson Conjecture states that every cubic bridgeless graph has six perfect matchings such that every edge of the graph is contained in exactly two of these perfect matchings.  ...  This lemma is further applied in the verification of Berge-Fulkerson Conjecture for some families of non-3-edge-colorable cubic graphs (such as, Goldberg snarks, flower snarks).  ...  Acknowledgements The first author was supported by the Natural Science Foundation of China (under the Grant Number 10871021).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2008.12.024">doi:10.1016/j.disc.2008.12.024</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vibu4orqwrbtndl7yr2binysoq">fatcat:vibu4orqwrbtndl7yr2binysoq</a> </span>
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Even Subdivision-Factors of Cubic Graphs [article]

Arthur Hoffmann-Ostenhof
<span title="2012-11-09">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As a consequence, we disprove a conjecture which was stated in an attempt to solve the circuit double cover conjecture.  ...  We show that any set of 2-connected graphs which is an even subdivison-factor of every 3-connected cubic graph, satisfies certain properties.  ...  Such matching M covering V (P ) corresponds to a perfect matching of P 10 . Hence, P − M consists of a path and a circuit C of length 5.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1211.1714v2">arXiv:1211.1714v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wkkxl5oeyfbotnovzsjwtouv6a">fatcat:wkkxl5oeyfbotnovzsjwtouv6a</a> </span>
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Graphs with the Circuit Cover Property

Brian Alspach, Luis Goddyn, Cun-Quan Zhang
<span title="">1994</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset.  ...  A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p .  ...  Sebö for stimulating discussion regarding the complexity of circuit covers.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170816123401/http://www.ams.org/journals/tran/1994-344-01/S0002-9947-1994-1181180-1/S0002-9947-1994-1181180-1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2c/f6/2cf6d084cf4d4b612138edfdf69c6b4be7852037.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2154711"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> jstor.org </button> </a>

Graphs with the circuit cover property

Brian Alspach, Luis Goddyn, Cun Quan Zhang
<span title="1994-01-01">1994</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/w3g32txdvneltemssag5nwfxcy" style="color: black;">Transactions of the American Mathematical Society</a> </i> &nbsp;
A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset.  ...  A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p .  ...  Sebö for stimulating discussion regarding the complexity of circuit covers.  ... 
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