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Packings and coverings of the complete directed multigraph with 3- and 4-circuits

F.E. Bennett, J. Yin
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In particular, we completely determine the packing number and covering number for the cases k --3 and 4.  ...  By a k-circuit we mean a directed cycle of length k. In this paper, we consider the problem of finding maximal packings and minimal coverings of 2DKv with k-circuits.  ...  Acknowledgements The first author would like to acknowledge the support of the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(95)00302-d">doi:10.1016/0012-365x(95)00302-d</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xe2ojzwzobbvzbpbtuueo4jrwi">fatcat:xe2ojzwzobbvzbpbtuueo4jrwi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170929210840/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/817/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDk1MDAzMDJk.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/5a/d2/5ad2a4bf6a18411ce541e98c8bdcb2db069653cd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(95)00302-d"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 58 of Mathematical Reviews Vol. , Issue 2000a [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Summary: “A packing of the complete directed symmetric graph DK, with m-circuits, denoted by (v,m)-DCP, is defined to be a family of arc-disjoint m-circuits of DK, such that any one arc of DK, occurs in  ...  COMBINATORICS 2000a:05157 05C70 05C20 05C38 Liang, Zhihe (PRC-HBN; Shijiazhuang) ; Kang, Qingde (PRC-HBN; Shijiazhuang) Packings of the complete directed graph with m-circuits.  ... 
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Page 5186 of Mathematical Reviews Vol. , Issue 96i [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(English summary) Discrete Math. 149 (1996), no. 1-3, 325-328. Summary: “A cycle packing in a [directed] multigraph is a ver- tex disjoint collection of [directed] elementary cycles.  ...  Summary: “We give a complete solution to the existence problem for complementary P3-decompositions of the complete multigraph, where P; denotes the path of length 3.” 96i:05129 05C70 05C65 Erdés, Paul  ... 
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Page 3436 of Mathematical Reviews Vol. , Issue 97F [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
E. (3-MSV; Halifax, NS); Yin, J. | Yin, Jian Xing] (PRC-SOO; Suzhou) Packings and coverings of the complete directed multigraph with 3- and 4-circuits.  ...  In particular, we completely determine the packing number and covering number for the cases k = 3 and 4.”  ... 
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Page 822 of Mathematical Reviews Vol. , Issue 2003B [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A maximum packing (minimum covering) with directed k-cycles of order n and index / is a directed k-cycle system of a graph formed from /D,, the directed multigraph on 7 vertices in which each pair of vertices  ...  Chris Rodger (1-ABRN-MS; Auburn, AL) 2003b:05043 05B30 05C38 05C70 Kang, Qingde (PRC-HNU; Shijiazhuang) ; Liang, Zhihe (PRC-HNU; Shijiazhuang) Optimal packings and coverings of 7D K,, with k-circuits.  ... 
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Page 1301 of Mathematical Reviews Vol. , Issue 91C [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Equivalently, the size of the minimum cardinality node-cover equals the maximum profit of a packing of edges and odd circuits, where the profit on an edge is 1 and the profit of a circuit of length 2k  ...  Summary: “A 1-factorization of the complete multigraph AK,, is called decomposable if some proper subset of the factors forms a 1-factorization of 4’K,, for some A’ < A; otherwise it is called inde- composable  ... 
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Matroid intersection, base packing and base covering for infinite matroids

Nathan Bowler, Johannes Carmesin
<span title="2014-09-04">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ujm6quky2va7pmzbitmgsrksbu" style="color: black;">Combinatorica</a> </i> &nbsp;
These include base packing, base covering, and matroid intersection and union.  ...  This new point of view also allows us to extend, and simplify the proofs of, some cases where these conjectures were known to be true.  ...  For the equivalence of the pairwise Packing/Covering conjecture with the Packing/Covering conjecture, we just need that M contains all the matroids U * 1,K and is closed under direct sums.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-014-2953-2">doi:10.1007/s00493-014-2953-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tf6nm2uxrzaf3kxayf4nt7evsa">fatcat:tf6nm2uxrzaf3kxayf4nt7evsa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808054237/http://www.math.uni-hamburg.de/spag/dm/papers/PackingCovering.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/34/e7/34e79eb07d8acc750553b2d812dcd322fb45971e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00493-014-2953-2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

A brief survey of perfect Mendelsohn packing and covering designs

F.E. Bennett
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
The values of the functions P(v; 3; ) and C(v; 3; ) have been determined completely for all v¿3 and ¿1.  ...  For k = 5 and = 1, the packing problem has been solved, with a few possible exceptions, but the covering problem remains to be investigated.  ...  In graph-theoretic terms, a (v; k; )-PMD is equivalent to a decomposition of the complete directed multigraph DK v on v vertices into k-circuits such that for any r; 16r6k−1, and for any two distinct vertices  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(99)00066-9">doi:10.1016/s0166-218x(99)00066-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/brn42a7qgnabpogvuvm6glra5y">fatcat:brn42a7qgnabpogvuvm6glra5y</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190417162306/https://core.ac.uk/download/pdf/81928752.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/da/7e/da7e2641f38dc44f2b5290a16ae2f5cbc79674e3.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(99)00066-9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 1535 of Mathematical Reviews Vol. 46, Issue 6 [page]

<span title="">1973</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Many of the known results on coloring the edges of un- directed graphs can be extended to multigraphs and to infinite graphs.  ...  every positive integer exactly once (=a complementing system); then «,/a, is an integer for some 1 4) in each of the following cases: (i) m=3 and m= 4; (ii) m=5 if all «, but one are integers; (iii) m2  ... 
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Treelike Snarks

Marién Abreu, Tomáš Kaiser, Domenico Labbate, Giuseppe Mazzuoccolo
<span title="2016-09-30">2016</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
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.  ...  By repeating the same process on each circuit of E(G) \ J 4 , we obtain a join J 1 such that {J 1 , J 2 , J 3 , J 4 } is a 2-packing and J 4 connected.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6008">doi:10.37236/6008</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zolqjvztd5gz3fpb6mz67xstay">fatcat:zolqjvztd5gz3fpb6mz67xstay</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180427065134/http://www.combinatorics.org/ojs/index.php/eljc/article/download/v23i3p54/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/d6/3d/d63d076c7a755cd6c577e39ae578ce85e4003122.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/6008"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Treelike snarks [article]

Marién Abreu, Tomas Kaiser, Domenico Labbate, Giuseppe Mazzuoccolo
<span title="2016-01-05">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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 Hagglund.  ...  In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number five and admit a 5-cycle double cover.  ...  of a graph is Conversely, suppose {J 1 , J 2 , J 3 , J 4 } a 2-packing of joins with J 4 connected.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1601.00870v1">arXiv:1601.00870v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s55n4cqdnrbt5hajdaro5uxrwq">fatcat:s55n4cqdnrbt5hajdaro5uxrwq</a> </span>
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On r-Simple k-Path and Related Problems Parameterized by k/r [article]

Gregory Gutin, Magnus Wahlström, Meirav Zehavi
<span title="2020-10-22">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In the (r,k)-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial P on n variables and integers k,r, decide whether P has a monomial of degree k where the degree of each  ...  In the p-Set (r,q)-Packing problem, given a universe V, positive integers p,q,r, and a collection H of sets of size p whose elements belong to V, decide whether there exists a subcollection H' of H of  ...  Moreover, by Properties 2 and 3 in Lemma 4, W has at most |X| walks with equal endpoints. Thus, the number of distinct walks we need to consider is ≤ 4|X| ≤ 4(|V (P, r)| + 2) ≤ 4(2k/r + 4).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1806.09108v2">arXiv:1806.09108v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hcu4slsv7rbcra5yt53qunx32a">fatcat:hcu4slsv7rbcra5yt53qunx32a</a> </span>
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Memory layout techniques for variables utilizing efficient DRAM access modes in embedded system design

Yoonseo Choi, Taewhan Kim, Hwansoo Han
<span title="">2005</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/rl7xk4fwazdrred2difr6v3lii" style="color: black;">IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems</a> </i> &nbsp;
From experiments with a set of benchmark programs, we confirm that our proposed techniques use on average 28.2% and 10.1% more page accesses and 82.9% and 107% more burst accesses than those by the order  ...  of first use and the technique of Panda et al.  ...  ( Step 3). The algorithm of finding a Euler circuit in a directed graph takes a time complexity of O(jE j).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcad.2004.837721">doi:10.1109/tcad.2004.837721</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2iksscd6efbotbhe7yrhpcoxia">fatcat:2iksscd6efbotbhe7yrhpcoxia</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922034705/http://koasas.kaist.ac.kr/bitstream/10203/476/1/TCAD2005.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/d0/bd/d0bd052aa8a9c40e65a65442e05dfe8cc4ff662c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tcad.2004.837721"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Combinatorial Optimization with Explicit Delineation of the Ground Set by a Collection of Subsets

Moshe Dror, James B. Orlin
<span title="">2008</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
The problems examined are generalizations of well known classical graph problems and include the minimal spanning tree, the assignment problem, a number of elementary machine scheduling problems, bin-packing  ...  We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund, 1997) given that the ground set of elements N has additional characteristics.  ...  The model (1),(2),and (4), is already NP-hard and inapproximable as a version of Set Cover with no further transformation.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/050636589">doi:10.1137/050636589</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pcmpjtuzfnel3o2rllmw3s5kkm">fatcat:pcmpjtuzfnel3o2rllmw3s5kkm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20100616184204/http://jorlin.scripts.mit.edu/docs/publications/107-CombOptWith.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/1d/d6/1dd6704bf3e43eb6fb0fe6e3e5efe4d82ae7aa53.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/050636589"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Page 766 of Mathematical Reviews Vol. 40, Issue 4 [page]

<span title="">1970</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Beineke, A survey of packings and coverings of graphs ; 1. Z. Bouwer, Section graphs for finite permutation groups ; D. W. Crowe, Nearly regular polyhedra with two exceptional faces; P.  ...  On the contrary, if anything, it will be directed more to those who have little or no acquaintance with computers rather than to the expert computer man.  ... 
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