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Sharp bounds for decompositions of graphs into completer-partite subgraphs

1996
*
Journal of Graph Theory
*

If G is a

doi:10.1002/(sici)1097-0118(199604)21:4<393::aid-jgt4>3.0.co;2-k
fatcat:75h3ajd3jzez7jz5j5mno2jgpi
*graph*on n vertices and*r*2 2, w e let m,(G) denote the minimum number*of**complete*multipartite*subgraphs*, with*r*or fewer parts, needed to*partition*the edge set, f(G). ... In determining m,(G), w e may assume that no two vertices*of*G have the same neighbor set. For such reduced*graphs*G, w e prove that m,(G) 2 log,(n + rl)/*r*. ...*COVERS**BY**COMPLETE**r*-*PARTITE**SUBGRAPHS*A*graph*G is determined*by*the n X k incidence matrix D*of*any*of*its*covers*(or decompositions)*by**complete**r*-*partite**subgraphs*: two vertices*of*G are adjacent if ...##
###
Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

2008
*
Graphs and Combinatorics
*

We classify the results into the following categories: vertex-

doi:10.1007/s00373-008-0789-5
fatcat:a4j64ccspze2td7cv3f247kf7u
*partitions**by*monochromatic*subgraphs*, such as cycles, paths, trees; vertex*partition**by*some kinds*of*heterochromatic*subgraphs*; the computational ... complexity*of*these*partition*problems; some kinds*of*large monochromatic and heterochromatic*subgraphs*. ... Conjecture 1 [42] The vertices*of*every*r*-edge colored*complete**graph*K n can be*covered**by*at most*r*vertex-disjoint monochromatic paths. ...##
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Finding Bipartite Partitions on Co-Chordal Graphs
[article]

2022
*
arXiv
*
pre-print

In this paper, we show that the biclique

arXiv:2203.02837v1
fatcat:j6sn6res3jcdjordlune6rnjm4
*partition*number (bp)*of*a co-chordal (complementary*graph**of*chordal)*graph*G = (V, E) is less than the number*of*maximal cliques (mc)*of*its complementary*graph*... Then, an O[|V|(|V|+|E^c|)]-time heuristic is proposed*by*applying lexicographic breadth-first search. Either heuristic gives us a biclique*partition**of*G with a size*of*mc(G^c)-1. ... Introduction The biclique*partition*number (bp)*of*a*graph*G is referred to as the least number*of**complete*bipartite (biclique)*subgraphs*that are required to*cover*the edges*of*the*graph*exactly once ...##
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Graphs with many r -cliques have large complete r -partite subgraphs

2008
*
Bulletin of the London Mathematical Society
*

We prove that for all

doi:10.1112/blms/bdm093
fatcat:5hjxqwywxfaxhj4cwltsksluwe
*r*≥2 and c>0, every*graph**of*order n with at least cn^*r*cliques*of*order*r*contains a*complete**r*-*partite**graph*with each part*of*size c^*r*n . ... This result implies a concise form*of*the Erdős-Stone theorem. ... Let H * be the*subgraph**of*H induced*by*the union*of*the members*of*S; clearly, H * = K*r*−1 (s, . . . , s). ...##
###
Partitioning 3-edge-coloured complete bipartite graphs into monochromatic cycles

2015
*
Electronic Notes in Discrete Mathematics
*

We show that any colouring with three colours

doi:10.1016/j.endm.2015.06.106
fatcat:4qf6yjrpxfastnoljsefmifgcm
*of*the edges*of*the*complete*bipartite*graph*K n,n contains 18 vertex-disjoint monochromatic cycles which together*cover*all vertices. ... The minimum number*of*cycles needed for such a*covering*is five, and we show that this lower bound is asymptotically true. This extends known results for*complete**graphs*. ... Introduction Cycle*partitioning**complete**graphs*Given an arbitrary colouring*of*the edges*of*a*graph*G with*r*colours, we are interested in determining the smallest number*of*monochromatic cycles that ...##
###
On Computing the Hamiltonian Index of Graphs
[article]

2019
*
arXiv
*
pre-print

The

arXiv:1912.01990v1
fatcat:z2ggfot3sbaphkrmvovbavtu5i
*r*-th iterated line*graph*L^*r*(G)*of*a*graph*G is defined*by*: (i) L^0(G) = G and (ii) L^*r*(G) = L(L^(*r*- 1)(G)) for*r*> 0, where L(G) denotes the line*graph**of*G. ... The NP-hard Eulerian Steiner*Subgraph*problem takes as input a*graph*G and a specified subset K*of*terminal vertices*of*G and asks if G has an Eulerian (that is: connected, and with all vertices*of*even ... Moreover, from Lemma 11 we get that P1 t P2 is the*partition**of*X defined*by*the*subgraph*Gt0 . 3. v? ∈ V(Gt0 ) holds, and V(Gt0 ) is a vertex*cover**of**graph*Gt . 4. ...##
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Page 2298 of Mathematical Reviews Vol. , Issue 87e
[page]

1987
*
Mathematical Reviews
*

The authors solve a problem suggested

*by*M. Tarsi*by*determining all minimum length cocycle*covers**of**complete**graphs*. ... The author finds all ordered 4-tuples (a,b,*r*,p) for which there exists a decomposition*of*a*complete**r*-*partite**graph**of*order p into two factors with diameters a and b. ...##
###
Covering the edges of a graph by three odd subgraphs

2006
*
Journal of Graph Theory
*

We prove that any finite simple

doi:10.1002/jgt.20170
fatcat:doh4bxbg75aj5af6wg7aksaka4
*graph*can be*covered**by*three*of*its odd*subgraphs*, and we construct an infinite sequence*of**graphs*where an edge-disjoint*covering**by*three odd*subgraphs*is not possible ... I also thank László Pyber for his advices and guidance in the domain*of**covering*problems. ... CCC 2 JOURNAL*OF**GRAPH*THEORY Theorem 2. Every finite simple*graph*can be*covered**by*three odd*subgraphs*. Example. ...##
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Vertex partitions of non-complete graphs into connected monochromatic k-regular graphs

2011
*
Discrete Mathematics
*

In a landmark paper, Erdős et al. (1991) [3] proved that if G is a

doi:10.1016/j.disc.2011.05.031
fatcat:qt4oh3cfkzgujlavsqtj3uvxj4
*complete**graph*whose edges are colored with*r*colors then the vertex set*of*G can be*partitioned*into at most cr 2 log*r*monochromatic ... Sárközy extended this result to non-*complete**graphs*, and Sárközy and Selkow extended it to k-regular*subgraphs*. ... The first author's research was supported in part*by*OTKA Grant No. K68322,*by*a János Bolyai Research Scholarship and*by*NSF Grant DMS-0968699. ...##
###
Page 1900 of Mathematical Reviews Vol. , Issue 89D
[page]

1989
*
Mathematical Reviews
*

A

*graph*G is divisible*by*¢ if its edge set can be*partitioned*into t subsets such that the*subgraphs*(called factors) induced*by*the subsets are all isomorphic. ... A G-decomposition*of*the*complete**graph*K,, is a*partition**of*its edge set into subsets such that each subset induces a*graph*isomorphic to G. ...##
###
An Improved Bound for Vertex Partitions by Connected Monochromatic K-Regular Graphs

2012
*
Journal of Graph Theory
*

*r*colors then the vertex set

*of*K n can be

*partitioned*into at most 100r log

*r*+ 2rk vertex disjoint connected monochromatic k-regular

*subgraphs*and vertices. ... Improving a result

*of*Sárközy and Selkow, we show that for all integers

*r*, k ≥ 2 there exists a constant n 0 = n 0 (

*r*, k ) such that if n ≥ n 0 and the edges

*of*the

*complete*

*graph*K n are colored with ... In every

*r*-coloring

*of*the edges

*of*a

*complete*

*graph*, its vertex set can be

*partitioned*into

*r*monochromatic cycles. ...

##
###
The achromatic number of a graph

1970
*
Journal of Combinatorial Theory
*

The concept

doi:10.1016/s0021-9800(70)80072-2
fatcat:tjph5jweyfgh7jza7qa5lxfb5u
*of*coloring a*graph*has been shown to be subsumed*by*that*of*an homomorphism. ... This led in [3] to the definition*of*a*complete*n-coloring*of*a*graph*G and suggested therefore a new invariant, which we now call the "achromatic number" 4,(G). ... V,,}*of*the set*of*points V*of*a*graph*G is a K-*partition**of*order m if every induced*subgraph*(Vi> is*complete*. ...##
###
Multipartite Ramsey numbers for odd cycles

2009
*
Journal of Graph Theory
*

We formulate the following conjecture: Let n ≥ 5 be an arbitrary positive odd integer; then, in any two-coloring

doi:10.1002/jgt.20364
fatcat:xqtavuut7jdqzijpqafaoavxne
*of*the edges*of*the*complete*5-*partite**graph*K((n−1)/2, (n−1)/2, (n−1)/2, (n−1)/2, 1) there ... is a monochromatic C n , a cycle*of*length n. ... One can easily*complete*the matching M*R*∪ M B to a matching M*covering*almost all vertices*of*V (H ) (here we can apply Lemma 4 again for the 2-colored*partite**graph*spanned*by*V (H )\(V (M*R*)∪ V (M ...##
###
Page 4118 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

A collection

*of*n*subgraphs**of*the*complete**graph*K, is an orthogonal double*cover*if every edge*of*the*complete**graph*lies in exactly two*of*the*subgraphs*, and any two*subgraphs*have exactly one common ... A &-*partition**of*a*graph*H is a*partition**of*V(H) into induced*subgraphs*isomorphic to members*of*the family ¥. ...##
###
Some complexity results about threshold graphs

1994
*
Discrete Applied Mathematics
*

The problem

doi:10.1016/0166-218x(94)90214-3
fatcat:62mrfm5cdfdepm2qtahxk4wp4y
*of*determining whether a*graph*G contains a threshold*subgraph*containing at least h edges is shown to be NP-*complete*if h is part*of*the input as the problems*of*minimum threshold*completion*... , weighted 2-threshold*partition*and weighted 2-threshold*covering*. ... This implies directly that the problem*of*the minimum*completion**of*a*graph*to obtain a threshold*graph*is NP-*complete*and we also show that the problems*of*finding a*partition*or a*covering**of*weight ...
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