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

2005
*
Discrete Mathematics
*

-fold

doi:10.1016/j.disc.2005.04.023
fatcat:wl6gjhsonrfkdgrxh3gsahkwmm
*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. ...##
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On some multigraph decomposition problems and their computational complexity

2004
*
Discrete Mathematics
*

The subject

doi:10.1016/j.disc.2003.07.005
fatcat:dreuozdsina4lktobjb2oidtta
*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*. ...##
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Multigraph decomposition into stars and into multistars

2005
*
Discrete Mathematics
*

We study the

doi:10.1016/j.disc.2005.03.012
fatcat:lr55eius5bafldqpayobxmzp6i
*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*? ...##
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Universal third parts of any complete 2-graph and none of DK_5

2013
*
Opuscula Mathematica
*

On the other hand, for each n ≥ 3 there is a universal third part F

doi:10.7494/opmath.2013.33.4.685
fatcat:3shuphanqbfjxpuwiqavtkzize
*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. ...##
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Multigraph decomposition into multigraphs with two underlying edges

2005
*
Discrete Mathematics & Theoretical Computer Science
*

International audience Due to some intractability considerations, reasonable formulation

doi:10.46298/dmtcs.3405
fatcat:r7x3fb7davcyzcl37hnid5jsbu
*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. ...##
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Triangle Decompositions of Planar Graphs
[article]

2015
*
arXiv
*
pre-print

A

arXiv:1504.00617v1
fatcat:2hlrv2f3cbdxdmgupwit4ri2cm
*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*...##
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Triangle decompositions of planar graphs

2016
*
Discussiones Mathematicae Graph Theory
*

A

doi:10.7151/dmgt.1882
fatcat:mrp7lpfs7rhjdmcbhtpousksna
*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*...##
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Decomposing Semi-complete Multigraphs and Directed Graphs into Paths of Length Two
[chapter]

2017
*
Lecture Notes in Computer Science
*

A P3-

doi:10.1007/978-3-319-53007-9_15
fatcat:2iulyi7wwzc3noegmvhaqwi6aq
*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. ...##
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The complexity of $P$4-decomposition of regular graphs and multigraphs

2015
*
Discrete Mathematics & Theoretical Computer Science
*

El-Zanati et al.(2014) showed that for all integers k ≥1, every 6k-regular

doi:10.46298/dmtcs.2128
fatcat:mnzr7byczjdlxmwxj6aee6isny
*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. ...##
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Hamilton decompositions of cartesian products of graphs

1991
*
Discrete Mathematics
*

We will extend some

doi:10.1016/0012-365x(91)90354-5
fatcat:5stki53e7fg2bd4qkr2jbmlmku
*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*. ...##
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A decomposition of Gallai multigraphs

2014
*
Discussiones Mathematicae Graph Theory
*

Moreover, we show that Gallai

doi:10.7151/dmgt.1740
fatcat:aui4bgh4ezazpmhnpp7hmbi3ie
*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. ...##
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DECOMPOSITIONS OF GENERALIZED COMPLETE GRAPHS

2009
*
Bulletin of the Australian Mathematical Society
*

In Chapter 6 we present a new technique-using edge labelled

doi:10.1017/s0004972709000793
fatcat:irrepaisd5htlhygdwf3dvgh6u
*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. ...##
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Line graphs of complete multipartite graphs have small cycle double covers

2002
*
Discrete Mathematics
*

Combin. 24 (2001) 91) that bridgeless line graphs

doi:10.1016/s0012-365x(02)00258-3
fatcat:2pimvy4lwbcyxoer6iezg7jqti
*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). ...##
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Decompositions of complete multigraphs into stars of varying sizes
[article]

2018
*
arXiv
*
pre-print

In 1979 Tarsi showed that an edge

arXiv:1807.10738v2
fatcat:fkcjpzo4lbbh3i7qzrpd65f2uu
*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*. ...##
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Cycle packings of the complete multigraph
[article]

2017
*
arXiv
*
pre-print

While cycle

arXiv:1701.05287v2
fatcat:n7es55rekbax3icil57rdkk3i4
*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. ...
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