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Packing cycles in graphs, II

Guoli Ding, Zhenzhen Xu, Wenan Zang
2003 Journal of combinatorial theory. Series B (Print)  
A graph G packs if for every induced subgraph H of G; the maximum number of vertexdisjoint cycles in H is equal to the minimum number of vertices whose deletion from H results in a forest.  ...  The purpose of this paper is to characterize all graphs that pack. r  ...  Let C be a maximum cycle packing in F : Then we define a cycle packing B in ðF \eÞ,P as follows: put B ¼ C if no cycle in C contains e; else, let C be the cycle in C containing e; and let % C be the cycle  ... 
doi:10.1016/s0095-8956(02)00007-2 fatcat:dp6xbwnqe5h7hpnxzpvo3mxsta

On maximum cycle packings in polyhedral graphs

Peter Recht, Stefan Stehling
2014 Electronic Journal of Graph Theory and Applications  
This paper addresses upper and lower bounds for the cardinality of a maximum vertex-/edgedisjoint cycle packing in a polyhedral graph G.  ...  Bounds on the cardinality of such packings are provided, that depend on the size, the order or the number of faces of G, respectively. Polyhedral graphs are constructed, that attain these bounds.  ...  In this paper, we investigate maximum cycle packings in polyhedral graphs G.  ... 
doi:10.5614/ejgta.2014.2.1.2 fatcat:sokv3td4jvdtvhadvftzxlr6ca

On the use of augmenting chains in chain packings

D. de Werra, F.S. Roberts
1991 Discrete Applied Mathematics  
In a graph G=(X,E), we assign to each node u a positive integer b(o)cdc(u), where C&(U) is the degree of v in G.  ...  We show that the use of augmenting chains fails in general to produce a new short chain packing from an old one, even for bipartite graphs, but that it does do so for the special case of trees.  ...  Acknowledgement This paper was written while the first author was visiting RUTCOR (Rutgers Center for Operations Research, Rutgers University) in May and June 1988. The support of the U.S.  ... 
doi:10.1016/0166-218x(91)90039-y fatcat:ipjdafgjcbg3fdrc7ozyh76iae

Between Proper and Strong Edge-Colorings of Subcubic Graphs [chapter]

Hervé Hocquard, Dimitri Lajou, Borut Lužar
2020 Lecture Notes in Computer Science  
Our research is motivated by the conjecture proposed in a recent paper on S-packing edgecolorings (N. Gastineau and O. Togni, On S-packing edge-colorings of cubic graphs, Discrete Appl.  ...  We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the conjecture for this class of graphs.  ...  This research has been done in the scope of the bilateral project between France and Slovenia, BI-FR/19-20-PROTEUS-001.  ... 
doi:10.1007/978-3-030-48966-3_27 fatcat:7uapml3a6vef3lmflby6kaj57m

Packing disjoint cycles over vertex cuts

Jochen Harant, Dieter Rautenbach, Peter Recht, Ingo Schiermeyer, Eva-Maria Sprengel
2010 Discrete Mathematics  
For a graph G, let ν(G) and ν (G) denote the maximum cardinalities of packings of vertexdisjoint and edge-disjoint cycles of G, respectively.  ...  We study the interplay of these two parameters and vertex cuts in graphs.  ...  Introduction Packing vertex-or edge-disjoint cycles in graphs is a very well-studied and classical graph-theoretical problem.  ... 
doi:10.1016/j.disc.2010.03.009 fatcat:xc53bpltzbedlmrrtbb26fli5q

Packing edge-disjoint cycles in graphs and the cyclomatic number

Jochen Harant, Dieter Rautenbach, Peter Recht, Friedrich Regen
2010 Discrete Mathematics  
For a graph G let µ(G) denote the cyclomatic number and let ν(G) denote the maximum number of edge-disjoint cycles of G.  ...  We prove that for every k ≥ 0 there is a finite set P (k) such that every 2-connected graph G for which µ(G) − ν(G) = k arises by applying a simple extension rule to a graph in P (k).  ...  For a cycle packing C, the set of edges contained in some cycle in C is denoted by E C . ν(G) ≤ µ(G) which holds for every graph G.  ... 
doi:10.1016/j.disc.2009.07.017 fatcat:pq2pth4hxjctfaoqbfefgap4zm

Page 5186 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
Summary: “A cycle packing in a [directed] multigraph is a ver- tex disjoint collection of [directed] elementary cycles.  ...  Summary: “Let G be a connected graph on n vertices, and let a, B, y and 6 be edge-disjoint cycles in G such that (i) a, B [resp. y, 6] are vertex-disjoint and (ii) ja|+|f| = |y| + |6| = n, where |a| denotes  ... 

Cycle packings of the complete multigraph [article]

Rosalind A. Cameron
2017 arXiv   pre-print
While cycle decompositions can give rise to packings by removing cycles from the decomposition, in general it is not known when there exists a packing of the complete multigraph with cycles of various  ...  In this paper we characterise exactly when there exists a packing of the complete multigraph with cycles of specified lengths m_1,m_2,...,m_τ.  ...  A cycle packing of a multigraph G is a packing P of G such that each submultigraph in P is a cycle.  ... 
arXiv:1701.05287v2 fatcat:n7es55rekbax3icil57rdkk3i4

On the Open Packing Number of a Graph

2020 International journal of recent technology and engineering  
A non-empty set of a graph G is an open packing set of G if no two vertices of S have a common neighbor in G.  ...  In this paper, the classes of trees and unicyclic graphs for which the value of is either 2 or 3 are characterized.  ...  Let C n be a cycle on 3  n vertices. Then II. MAIN RESULTS In this section we characterize the connected graphs for which the value of the open packing number is 2.  ... 
doi:10.35940/ijrte.d1038.1284s419 fatcat:xcwejf4j4rf25m2hyzqimwmvlm

Maximum cycle packing in Eulerian graphs using local traces

Peter Recht, Eva-Maria Sprengel
2015 Discussiones Mathematicae Graph Theory  
We prove that every maximum edge-disjoint cycle packing Z * of G induces a maximum trace T (v) at v for every v ∈ V .  ...  Moreover, if G is Eulerian then sufficient conditions are given that guarantee that the sets of cycles inducing maximum local traces of G also induce a maximum cycle packing of G.  ...  Packing edge-disjoint cycles in graphs is a classical graph-theoretical problem.  ... 
doi:10.7151/dmgt.1785 fatcat:6eo572esg5b55c4yd64tba4mxq

Set packing relaxations of some integer programs

Ralf Borndörfer, Robert Weismantel
2000 Mathematical programming  
(ii) If C is an odd cycle in G, the odd cycle inequality i∈C x i ≤ (|C| − 1)/2 is valid for P SSP .  ...  Then: (i) Orthonormal representation inequalities violated by x can be separated in polynomial time. (ii) Odd cycle inequalities violated by x can be separated in polynomial time.  ...  cycle inequality in the link graph if and only if a violated cycle of cycles inequality in G exists  ... 
doi:10.1007/pl00011381 fatcat:p7rw7e5i7fgill3sxzmq3fem2y

Packing (2^k+1-1)-order perfect binary trees into (k+1)-connected graph [article]

Jia Zhao, Jianfeng Guan, Changqiao Xu, Hongke Zhang
2013 arXiv   pre-print
We also prove that the T_k packing problem in an arbitrary graph is NP-hard, and propose the distributed approximation algorithms.  ...  Let G=(V,E) and H be two graphs. Packing problem is to find in G the largest number of independent subgraphs each of which is isomorphic to H. Let U⊂V.  ...  According to the property of 2-connected graphs [2] , We can reconstruct the 2-connected graph G in this way: (i) to find in G a cycle C; (ii)to add C-paths to C to form a graph H; (iii) to add H-paths  ... 
arXiv:1309.3825v1 fatcat:qb6uyasudvdblbxafhk3pw3ede

Jones' Conjecture in subcubic graphs [article]

Marthe Bonamy, François Dross, Tomáš Masařík, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk
2019 arXiv   pre-print
We confirm Jones' Conjecture for subcubic graphs. Namely, if a subcubic planar graph does not contain k+1 vertex-disjoint cycles, then it suffices to delete 2k vertices to obtain a forest.  ...  A cycle packing of a (multi)graph G is a set of vertex disjoint cycles that appear in G as subgraphs. We denote the maximum size of a cycle packing of G by cp(G).  ...  Then for i ∈ {1, 2}, in every cycle packing of G AB i there is a cycle containing e A and e B .  ... 
arXiv:1912.01570v1 fatcat:tg2tfy7pzzfqhetloigzl34grq

Maximum packings with odd cycles

Saad I. El-Zanati
1994 Discrete Mathematics  
We show how to obtain maximum packings of K", +v with k-cycles when k>3 is odd, g a positive integer, and v even with O<v< 2k.  ...  Moreover, under certain conditions on v, we obtain maximum packings of Kzkg+".  ...  ii Fi. This decomposition is a maximum packing of K,,,+ u since its leave is a l-factor in KZkg. 0 Hence, 2kg+u is k-nice. 0 Corollary 4.5.  ... 
doi:10.1016/0012-365x(94)90375-1 fatcat:3vqkhl3x55fbdc4rhcmfilts2e

Decomposing various graphs into short even-length cycles [article]

Daniel Horsley
2012 arXiv   pre-print
We then use this result to obtain results on incomplete even cycle systems with a hole and on decompositions of complete multipartite graphs into cycles of uniform even length.  ...  We prove that a complete bipartite graph can be decomposed into cycles of arbitrary specified lengths provided that the obvious necessary conditions are satisfied, the length of each cycle is at most the  ...  Acknowledgement This research was supported in part by the Atlantic Association for Research in the Mathematical Sciences.  ... 
arXiv:1204.3368v1 fatcat:gtbdsrvttnbmrfyrsadrzcsxta
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