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Decompositions of Edge-Colored Complete Graphs

2000
*
Journal of combinatorial theory. Series A
*

We prove an asymptotic existence theorem for

doi:10.1006/jcta.1999.3005
fatcat:z3hprbnb7fhnnj2nxkjtkurgzu
*decompositions**of**edge*-*colored**complete**graphs*into prespecified*edge*-*colored*subgraphs. Many combinatorial design problems fall within this framework. ... Applications*of*our main theorem require calculations involving the numbers*of**edges**of*each*color*and degrees*of*each*color*class*of**edges*for the*graphs*allowed in the*decomposition*. ... ACKNOWLEDGMENT The work*of*the second author was supported in part by NSF Grant DMS-8703898. ...##
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Decompositions of edge-colored infinite complete graphs into monochromatic paths
[article]

2016
*
arXiv
*
pre-print

infinite

arXiv:1502.04955v2
fatcat:ixzaixozync7lkclbzvytpa4vy
*complete**graph*can be partitioned into 4 monochromatic squares*of*paths, but not necessarily into 3, (4.) the vertex set*of*every 2-*edge**colored**complete**graph*on ω_1 can be partitioned into 2 ... An r-*edge**coloring**of*a*graph*or hypergraph G=(V,E) is a map c:E→{0, ..., r-1}. ... The third author was partially supported by the Ontario Trillium Fellowship at the University*of*Toronto. ...##
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More on decompositions of edge-colored complete graphs

2008
*
Discrete Mathematics
*

K (r) n will denote the

doi:10.1016/j.disc.2007.08.044
fatcat:r3zchza2wfe2lmzsg6gs2wp4k4
*complete**graph*with r*edges*joining any pair*of*distinct vertices, one*of*each*of*the r*colors*. ... Let G be a family*of**graphs*whose*edges*are*colored*with elements from a set R*of*r*colors*. We assume no two vertices*of*G are joined by more than one*edge**of**color*i for any i ∈ R, for each G ∈ G. ... Work on balanced*decompositions*and*decompositions**of*directed*edge*-*colored**complete**graphs*is in progress and will appear elsewhere. ...##
###
Decompositions of edge-colored infinite complete graphs into monochromatic paths

2017
*
Discrete Mathematics
*

, v 1 , . . .

doi:10.1016/j.disc.2016.09.028
fatcat:y6tj72a7b5helcqe337ufgcnjq
*of*distinct vertices such that |i − j| ≤ k implies that {v i , v j } is an*edge*), • every 2-*edge**colored**complete**graph*on N can be partitioned into 4 monochromatic squares*of*paths, but ... not necessarily into 3, • every 2-*edge**colored**complete**graph*on ω 1 can be partitioned into 2 monochromatic paths with distinct*colors*. ... Monochromatic path*decompositions**of*K ω 1 The aim*of*this section is to extend Rado's Theorem 2.5 to 2-*edge**colored**complete**graphs**of*size ω 1 . ...##
###
Multicolored forests in bipartite decompositions of graphs

1991
*
Journal of combinatorial theory. Series B (Print)
*

A clique

doi:10.1016/0095-8956(91)90059-s
fatcat:7xsvrenz2rbcrpiwyyfvppahlu
*decomposition**of*a*graph*G is an*edge*-*coloring**of*G such that each*color*class is the set*of*all*edges**of*a*complete*subgraph*of*G. ... A bipartite*decomposition**of*a*graph*G is an*edge*-*coloring**of*G such that each*color*class is the set*of*all*edges**of*a*complete*bipartite subgraph*of*G. ...##
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Page 778 of Mathematical Reviews Vol. , Issue 2001B
[page]

2001
*
Mathematical Reviews
*

, a

*complete**edge*-r-*colored*digraph has a*decomposition*into isomorphic copies*of*G. ... This paper deals with*edge**decompositions**of**complete*r-*colored*directed*graphs*(in such*graphs*, for every two different vertices x and y and for every*color*c in the fixed set*of*r possible*colors*, there ...##
###
Research Problems from the Fourth Cracow Conference (Czorsztyn, 2002)

2007
*
Discrete Mathematics
*

If so, is it true that, for sufficiently large n, every (n − 1)-circuit

doi:10.1016/j.disc.2006.09.039
fatcat:r2z7zf5vmjdu5pb2gmbgpo33wi
*decomposition**of*it is nomadic? A tournament is a digraph obtained by orienting the*edges**of*a*complete**graph*. ... References A natural generalization for*graphs*is defined as follows. A*coloring**of*the*edges**of*a*graph*G is nonrepetitive if along each path the sequence*of**colors*is nonrepetitive. ... It was conjectured in [1] that when n is sufficiently large in terms*of*k, every simple n-vertex*graph*G with more than 3 2 k − 2 (n − k + 1)*edges*has a k-connected subgraph (for k 7, it may suffice ...##
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Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications
[article]

2018
*
arXiv
*
pre-print

Also, a 1-in-Degree

arXiv:1801.04472v1
fatcat:xed6wahch5fx5op7zqu5aiydnu
*decomposition**of*a*graph*G is a*decomposition**of*the vertices*of*G into two parts A and B such that each vertex in the*graph*G has exactly one neighbor in part A. ... A Not-All-Equal (NAE)*decomposition**of*a*graph*G is a*decomposition**of*the vertices*of*G into two parts such that each vertex in G has at least one neighbor in each part. ... Acknowledgments The authors would like to thank the anonymous referees for their useful comments which helped to improve the presentation*of*this paper. ...##
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Optimization of the Functional Decomposition of Parallel and Distributed Computations in Graph Coloring With the Use of High-Performance Computing

2022
*
IEEE Access
*

These computations contain large number

doi:10.1109/access.2022.3162215
fatcat:67bqxac3r5hatg7x6ufsvgf73q
*of*calls*of*sequential, recursive algorithm for NP-*complete*problem -proper*edge**coloring**of**graph*. ... This article presents methods for correct*decomposition*for high performance computations related to large sets*of**graphs*. ...*EDGE**COLORING**OF*SNARKS The problem*of*proper*edge**coloring**of*cubic*graphs*, described in this section*of*the paper, is classified as an NP-*complete*problem [16] . ...##
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On intervalizing k-colored graphs for DNA physical mapping

1996
*
Discrete Applied Mathematics
*

The problem

doi:10.1016/s0166-218x(96)00057-1
fatcat:uvjh5zrd7zfcnjod3opddsqjni
*of*determining whether a given k-*colored**graph*is a subgraph*of*a properly*colored*interval*graph*has an application in DNA physical mapping. ... In this paper, we study the problem for the case that the number*of**colors*k is fixed. ... . , Vl) is a proper path*decomposition**of*G. Note that PD has width k -1. Then the interval*completion**of*G for PD is a properly k-*colored*interval*graph*. Proof. ...##
###
Complete Edge-Colored Permutation Graphs
[article]

2020
*
arXiv
*
pre-print

Using the modular

arXiv:2004.07118v1
fatcat:rsatvgun7zhqzoin33lwyc5qoi
*decomposition*as a framework we demonstrate that*complete**edge*-*colored*permutation*graphs*are characterized in terms*of*their strong prime modules, which induce also*complete**edge*-*colored*... We introduce the concept*of**complete**edge*-*colored*permutation*graphs*as*complete**graphs*that are the*edge*-disjoint union*of*"classical" permutation*graphs*. ... Acknowledgements This work was funded in part by the German Research Foundation (DFG STA 850/49-1) and Federal Ministry*of*Education and Research (BMBF 031L0164C). ...##
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On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions
[article]

2015
*
arXiv
*
pre-print

Symbolic ultrametrics define

arXiv:1501.03931v2
fatcat:qmtkxueapzenrjwue4gyd66z6u
*edge*-*colored**complete**graphs*K_n and yield a simple tree representation*of*K_n. ... An upper bound for the integer k is derived and it is shown that determining whether a*graph*has a cograph 2-*decomposition*, resp., 2-partition is NP-*complete*. ... = G m (δ) = (V, E m ) with*edge*set E m = {{x, y} | δ(x, y) = m, x, y ∈ V } . (2) Thus, the map δ can be considered as an*edge**coloring**of*a*complete**graph*K |V | , where each*edge*[x, y] obtains*color*...##
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On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions
[chapter]

2015
*
Lecture Notes in Computer Science
*

Symbolic ultrametrics define

doi:10.1007/978-3-319-21398-9_48
fatcat:p52o4svj5fbl7mppu74jylvt7y
*edge*-*colored**complete**graphs*Kn and yield a simple tree representation*of*Kn. ... An upper bound for the integer k is derived and it is shown that determining whether a*graph*has a cograph 2-*decomposition*, resp., 2-partition is NP-*complete*. ... = G m (δ) = (V, E m ) with*edge*set E m = {{x, y} | δ(x, y) = m, x, y ∈ V } . (2) Thus, the map δ can be considered as an*edge**coloring**of*a*complete**graph*K |V | , where each*edge*[x, y] obtains*color*...##
###
Parallel comparability graph recognition and modular decomposition
[chapter]

1996
*
Lecture Notes in Computer Science
*

n; m and respectively denote the number

doi:10.1007/3-540-60922-9_15
fatcat:noxg7zk23bef3loditjjxaveta
*of*vertices, the number*of**edges*and the maximal degree*of*the undirected input*graph*). ... Parallel algorithms for nding a transitive orientation and the modular*decomposition**of*any undirected*graph*are deduced from an extension*of*the theory*of*Golumbic toward modular*decomposition*. ... (ii) The*graph*with the supervertices and the superedges*of*M as vertices and*edges*is a*complete**graph*, seeFigure 5. ...##
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Decomposition by clique separators

1985
*
Discrete Mathematics
*

We give an O(nm)-time algorithm for finding a

doi:10.1016/0012-365x(85)90051-2
fatcat:en7gsgzgxvdz3ddba7ay75mbnu
*decomposition**of*an n-vertex, m-*edge**graph*. ... We consider the problem*of*decomposing a*graph*by means*of*clique separators, by which we mean finding cliques (*complete**graphs*) whose removal disconnects the*graph*. ... Acknowledgment My thanks to Marty Golumbic, whose lecture on EPT*graphs*stimulated this research. ...
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