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Decompositions of Edge-Colored Complete Graphs

Esther R. Lamken, Richard M. Wilson
2000 Journal of combinatorial theory. Series A  
We prove an asymptotic existence theorem for 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.  ... 
doi:10.1006/jcta.1999.3005 fatcat:z3hprbnb7fhnnj2nxkjtkurgzu

Decompositions of edge-colored infinite complete graphs into monochromatic paths [article]

M. Elekes, D. T. Soukup, L. Soukup, Z. Szentmiklóssy
2016 arXiv   pre-print
infinite 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.  ... 
arXiv:1502.04955v2 fatcat:ixzaixozync7lkclbzvytpa4vy

More on decompositions of edge-colored complete graphs

Anna Draganova, Yukiyasu Mutoh, Richard M. Wilson
2008 Discrete Mathematics  
K (r) n will denote the 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.  ... 
doi:10.1016/j.disc.2007.08.044 fatcat:r3zchza2wfe2lmzsg6gs2wp4k4

Decompositions of edge-colored infinite complete graphs into monochromatic paths

Márton Elekes, Dániel T. Soukup, Lajos Soukup, Zoltán Szentmiklóssy
2017 Discrete Mathematics  
, v 1 , . . . 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 .  ... 
doi:10.1016/j.disc.2016.09.028 fatcat:y6tj72a7b5helcqe337ufgcnjq

Multicolored forests in bipartite decompositions of graphs

Noga Alon, Richard A Brualdi, Bryan L Shader
1991 Journal of combinatorial theory. Series B (Print)  
A clique 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.  ... 
doi:10.1016/0095-8956(91)90059-s fatcat:7xsvrenz2rbcrpiwyyfvppahlu

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)

Rafał Kalinowski, Mariusz Meszka
2007 Discrete Mathematics  
If so, is it true that, for sufficiently large n, every (n − 1)-circuit 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  ... 
doi:10.1016/j.disc.2006.09.039 fatcat:r2z7zf5vmjdu5pb2gmbgpo33wi

Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications [article]

Ali Dehghan and Mohammad-Reza Sadeghi and Arash Ahadi
2018 arXiv   pre-print
Also, a 1-in-Degree 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.  ... 
arXiv:1801.04472v1 fatcat:xed6wahch5fx5op7zqu5aiydnu

Optimization of the Functional Decomposition of Parallel and Distributed Computations in Graph Coloring With the Use of High-Performance Computing

Jarmila Skrinarova, Adam Dudas
2022 IEEE Access  
These computations contain large number 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] .  ... 
doi:10.1109/access.2022.3162215 fatcat:67bqxac3r5hatg7x6ufsvgf73q

On intervalizing k-colored graphs for DNA physical mapping

Hans L. Bodlaender, Babette de Fluiter
1996 Discrete Applied Mathematics  
The problem 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.  ... 
doi:10.1016/s0166-218x(96)00057-1 fatcat:uvjh5zrd7zfcnjod3opddsqjni

Complete Edge-Colored Permutation Graphs [article]

Tom Hartmann, Max Bannach, Martin Middendorf, Peter F. Stadler, Nicolas Wieseke, Marc Hellmuth
2020 arXiv   pre-print
Using the modular 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).  ... 
arXiv:2004.07118v1 fatcat:rsatvgun7zhqzoin33lwyc5qoi

On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions [article]

Marc Hellmuth, Nicolas Wieseke
2015 arXiv   pre-print
Symbolic ultrametrics define 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  ... 
arXiv:1501.03931v2 fatcat:qmtkxueapzenrjwue4gyd66z6u

On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions [chapter]

Marc Hellmuth, Nicolas Wieseke
2015 Lecture Notes in Computer Science  
Symbolic ultrametrics define 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  ... 
doi:10.1007/978-3-319-21398-9_48 fatcat:p52o4svj5fbl7mppu74jylvt7y

Parallel comparability graph recognition and modular decomposition [chapter]

Michel Morvan, Laurent Viennot
1996 Lecture Notes in Computer Science  
n; m and respectively denote the number 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.  ... 
doi:10.1007/3-540-60922-9_15 fatcat:noxg7zk23bef3loditjjxaveta

Decomposition by clique separators

Robert E. Tarjan
1985 Discrete Mathematics  
We give an O(nm)-time algorithm for finding a 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.  ... 
doi:10.1016/0012-365x(85)90051-2 fatcat:en7gsgzgxvdz3ddba7ay75mbnu
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