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Balanced star decompositions of regular multigraphs and λ-fold complete bipartite graphs

Hung-Chih Lee, Chiang Lin
2005 Discrete Mathematics  
-fold complete bipartite graph K m,n to have a balanced S k -decomposition.  ...  A balanced S k -decomposition of a multigraph G is a family D of subgraphs of G whose edge sets form a partition of the edge set of G such that each member of D is isomorphic to S k , and every vertex  ...  Acknowledgements The authors thank the referees for their comments which improved the readability of this paper.  ... 
doi:10.1016/j.disc.2005.04.023 fatcat:wl6gjhsonrfkdgrxh3gsahkwmm

On some multigraph decomposition problems and their computational complexity

Miri Priesler (Moreno), Michael Tarsi
2004 Discrete Mathematics  
The subject of this article is the complexity of multigraph decomposition, that is the case where multiple edges are allowed.  ...  The H -decomposition computational problem is deÿned as follows: Given an input graph G, can its edge set be partitioned into isomorphic copies of H ?  ...  Deciding whether an input graph admits such a partition is NPC by Theorem 2.4. Theorem 2.6. S 1; 2 -5 decomposition is NP-complete.  ... 
doi:10.1016/j.disc.2003.07.005 fatcat:dreuozdsina4lktobjb2oidtta

Multigraph decomposition into stars and into multistars

Miri Priesler (Moreno), Michael Tarsi
2005 Discrete Mathematics  
We study the decomposition of multigraphs with a constant edge multiplicity into copies of a fixed star H = K 1,t : We present necessary and sufficient conditions for such a decomposition to exist where  ...  t = 2 and prove NP-completeness of the corresponding decision problem for any t 3.  ...  : Does an input multigraph M admit an H-decomposition?  ... 
doi:10.1016/j.disc.2005.03.012 fatcat:lr55eius5bafldqpayobxmzp6i

Universal third parts of any complete 2-graph and none of DK_5

Artur Fortuna, Zdzisław Skupień
2013 Opuscula Mathematica  
On the other hand, for each n ≥ 3 there is a universal third part F of the complete 2-graph 2 Kn on n vertices, i.e., for each edge subset R of size |R| = 2 Kn mod 3, there is an F -decomposition of 2  ...  Nevertheless, none of their orientations is a universal third part of the corresponding complete digraph.  ...  Acknowledgments The research of the first author was partially supported by the Polish Ministry of Science and Higher Education under grant N N201 391137.  ... 
doi:10.7494/opmath.2013.33.4.685 fatcat:3shuphanqbfjxpuwiqavtkzize

Multigraph decomposition into multigraphs with two underlying edges

Miri Priesler, Michael Tarsi
2005 Discrete Mathematics & Theoretical Computer Science  
International audience Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph  ...  We present necessary and sufficient conditions for H-decomposability of G, which hold when certain size parameters of G lies within some bounds which depends on the multiplicities of the two edges of H  ...  Extended Abstract Given two graphs H and G, an H-decomposition of G is a partition of the edge set of G into disjoint isomorphic copies of H.  ... 
doi:10.46298/dmtcs.3405 fatcat:r7x3fb7davcyzcl37hnid5jsbu

Triangle Decompositions of Planar Graphs [article]

Christina Mynhardt, Christopher van Bommel
2015 arXiv   pre-print
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational  ...  Finally, if G is a multigraph with the complete graph of order 4 as underlying graph, we give necessary and sufficient conditions on the multiplicities of its edges for G to be triangle and rationally  ...  A K 3 -decomposition of G is obtained by first deleting uvw, partitioning the faces into two sets so that one set contains only triangles, which form part of the decomposition, and reinserting uvw to complete  ... 
arXiv:1504.00617v1 fatcat:2hlrv2f3cbdxdmgupwit4ri2cm

Triangle decompositions of planar graphs

Christina M. Mynhardt, Christopher M. van Bommel
2016 Discussiones Mathematicae Graph Theory  
A multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational  ...  We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0, 1 and 1 2 .  ...  A K 3 -decomposition of G is obtained by first deleting uvw, partitioning the faces into two sets so that one set contains only triangles, which form part of the decomposition, and reinserting uvw to complete  ... 
doi:10.7151/dmgt.1882 fatcat:mrp7lpfs7rhjdmcbhtpousksna

Decomposing Semi-complete Multigraphs and Directed Graphs into Paths of Length Two [chapter]

Ajit A. Diwan, Sai Sandeep
2017 Lecture Notes in Computer Science  
A P3-decomposition of a graph is a partition of the edges of the graph into paths of length two.  ...  We give a necessary and sufficient condition for a semi-complete multigraph, that is a multigraph with at least one edge between each pair of vertices, to have a P3-decomposition.  ...  Let G be a semi-complete multigraph of even size and let M be a matching with maximum weight in G. G has a P 3 -decomposition iff for every subset of edges S ⊆ M , w(N (S)) ≥ w(S). Proof.  ... 
doi:10.1007/978-3-319-53007-9_15 fatcat:2iulyi7wwzc3noegmvhaqwi6aq

The complexity of $P$4-decomposition of regular graphs and multigraphs

Ajit Diwan, Justine Dion, David Mendell, Michael Plantholt, Shailesh Tipnis
2015 Discrete Mathematics & Theoretical Computer Science  
El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular multigraph with µ(G) ≤2k has a P4-decomposition.  ...  We prove that for all integers k ≥2, the problem of determining if P4 decomposes a (2k + 1)-regular graph is NP-Complete.  ...  Acknowledgements We would like to thank the anonymous referee for a careful reading of our paper and in particular for pointing out an error in our original proof of Theorem 11.  ... 
doi:10.46298/dmtcs.2128 fatcat:mnzr7byczjdlxmwxj6aee6isny

Hamilton decompositions of cartesian products of graphs

Richard Stong
1991 Discrete Mathematics  
We will extend some of our results to multigraphs.  ...  ., Hamilton decompositions of Cartesian products of graphs, Discrete Mathematics 90 (1991) 169-190. Let G1 and G2 be graphs that are decomposable into Hamilton cycles.  ...  If C', . . . , C" is a decomposition of a multigraph into Hamilton cycles, an m-matching or m-semimatching M is said to be orthogonal to the decomposition if (M rl C'l = 1 for i = 1, . . . , m.  ... 
doi:10.1016/0012-365x(91)90354-5 fatcat:5stki53e7fg2bd4qkr2jbmlmku

A decomposition of Gallai multigraphs

Alexander Halperin, Colton Magnant, Kyle Pula
2014 Discussiones Mathematicae Graph Theory  
Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees.  ...  A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles.  ...  Acknowledgment The authors would like to thank the anonymous referees for helpful suggestions, including the present form of the proof of Lemma 3.  ... 
doi:10.7151/dmgt.1740 fatcat:aui4bgh4ezazpmhnpp7hmbi3ie

DECOMPOSITIONS OF GENERALIZED COMPLETE GRAPHS

BENJAMIN R. SMITH
2009 Bulletin of the Australian Mathematical Society  
In Chapter 6 we present a new technique-using edge labelled decompositions of multigraphs-for obtaining k-cycle decompositions of K n * K m in cases where k is a factor of m 2 .  ...  A decomposition of a graph G is a collection of edge-disjoint subgraphs of G whose edges partition the edges of G.  ... 
doi:10.1017/s0004972709000793 fatcat:irrepaisd5htlhygdwf3dvgh6u

Line graphs of complete multipartite graphs have small cycle double covers

J.M. Fish, R. Klimmek, K. Seyffarth
2002 Discrete Mathematics  
Combin. 24 (2001) 91) that bridgeless line graphs of complete graphs, complete bipartite graphs, and planar graphs have small cycle double covers.  ...  In this paper, we extend the result for complete bipartite graphs, and show that the line graph of any complete multipartite graph (other than K1;2) has a small cycle double cover.  ...  The associated multigraph, M P (x), of a vertex x ∈V (G), is a multigraph on the vertices of N (x); endpoints vx and ux of a path in P(x) give rise to the edge vu in the associated multigraph M P (x).  ... 
doi:10.1016/s0012-365x(02)00258-3 fatcat:2pimvy4lwbcyxoer6iezg7jqti

Decompositions of complete multigraphs into stars of varying sizes [article]

Rosalind A. Cameron, Daniel Horsley
2018 arXiv   pre-print
In 1979 Tarsi showed that an edge decomposition of a complete multigraph into stars of size m exists whenever the obvious necessary conditions hold.  ...  We show that the general problem of when a complete multigraph admits a decomposition into stars of sizes m_1,...  ...  Thanks to Ramin Javadi for pointing out an error in a previous version of the proof of NP-completeness.  ... 
arXiv:1807.10738v2 fatcat:fkcjpzo4lbbh3i7qzrpd65f2uu

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.  ...  Acknowledgements The author was supported by a Monash University Faculty of Science Postgraduate Publication Award.  ... 
arXiv:1701.05287v2 fatcat:n7es55rekbax3icil57rdkk3i4
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