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Complete bipartite graphs with no rainbow paths

2016
*
International Journal of Contemporary Mathematical Sciences
*

In this paper, we consider the edge colorings of

doi:10.12988/ijcms.2016.6951
fatcat:7layqmcvrjc7pmmirleo3guk24
*complete*bipartite*graphs*that contain*no**rainbow**path*P t . ... Motivated by questions in Ramsey theory, Thomason and Wagner described the edge colorings of*complete**graphs*that contain*no**rainbow**path*P t of order t. ... Introduction Motivated by questions in Ramsey theory, Thomason and Wagner [7] considered the edge colorings of*complete**graphs*that contain*no**rainbow**path*P t of order t. ...##
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Note on the complexity of deciding the rainbow connectedness for bipartite graphs
[article]

2011
*
arXiv
*
pre-print

A

arXiv:1109.5534v2
fatcat:lkfy6t6zb5hedbwrrq2ezvrny4
*path*in an edge-colored*graph*is said to be a*rainbow**path*if*no*two edges on the*path*have the same color. ... Moreover, it is known that deciding whether a given edge-colored (*with*an unbound number of colors)*graph*is*rainbow*connected is NP-*Complete*. ... A u − v*path*P in G is a*rainbow**path*if*no*two edges of P are colored the same. ...##
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Complexity results for rainbow matchings

2014
*
Theoretical Computer Science
*

The latter is somewhat surprising, since, to the best of our knowledge,

doi:10.1016/j.tcs.2013.12.013
fatcat:inezqohlejgedn433llc72fuvm
*no*(unweighted)*graph*problem prior to our result is known to be NP-hard for simple*paths*. ... A*rainbow*matching in an edge-colored*graph*is a matching whose edges have distinct colors. ... Corollary 14.*rainbow*matching is NP-*complete*, even when restricted to one of the following classes of edge-colored*graphs*. 1.*Complete**graphs*. Properly edge-colored*paths*. 3. ...##
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Rainbow connections for planar graphs and line graphs
[article]

2011
*
arXiv
*
pre-print

We also give upper bounds of the

arXiv:1110.3147v2
fatcat:fsizn4hurndh7gg5myhjf7gvcy
*rainbow*connection number of outerplanar*graphs**with*small diameters. ... An edge-colored*graph*G is*rainbow*connected if any two vertices are connected by a*path*whose edges have distinct colors. ... In fact it is already NP-*complete*to decide whether rc(G) = 2, and in fact it is already NP-*complete*to decide whether a given edge-colored (*with*an unbounded number of colors)*graph*is*rainbow*connected ...##
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Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
[article]

2018
*
arXiv
*
pre-print

A

arXiv:1405.6893v7
fatcat:gobg6k74yreybbn5s3hruy6gbm
*path*in an edge-colored*graph*G is*rainbow*if*no*two edges of it are colored the same. The*graph*G is*rainbow*-connected if there is a*rainbow**path*between every pair of vertices. ... Furthermore, there exists*no*polynomial-time algorithm for approximating the strong*rainbow*connection number of an n-vertex split*graph**with*a factor of n^1/2-ϵ for any ϵ > 0 unless P = NP. ... Introduction Let G be an edge-colored undirected*graph*that is simple and finite. A*path*in G is*rainbow*if*no*two edges of it are colored the same. ...##
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Further hardness results on rainbow and strong rainbow connectivity

2016
*
Discrete Applied Mathematics
*

A

doi:10.1016/j.dam.2015.07.041
fatcat:brax3buo7jbwzgfvwf2luyrqqy
*path*in an edge-colored*graph*is*rainbow*if*no*two edges of it are colored the same. The*graph*is said to be*rainbow*connected if there is a*rainbow**path*between every pair of vertices. ... We show that for block*graphs*, which form a subclass of chordal*graphs*,*Rainbow*connectivity is*complete*while Strong*rainbow*connectivity is in . ... is*no*polynomial time algorithm to*rainbow*color*graphs**with*less than twice the optimum number of colors, unless P = NP. Computing the strong*rainbow*connection number is known to be hard as well. ...##
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The rainbow connection of windmill and corona graph

2014
*
Applied Mathematical Sciences
*

In this paper we determine the exact values of the windmill

doi:10.12988/ams.2014.48632
fatcat:to6swntqxvgknabze6jpdhhlyy
*graph*K (n) m . Moreover, we compute the rc(G • H) where G or H is*complete**graph*K m or*path*P 2*with*m is an integer. ... The*rainbow*connection number of G, denoted by rc(G), is the smallest number of colors needed to color its edges, so that every pair of its vertices is connected by at least one*path*in which*no*two edges ... A*path*is*rainbow*if*no*two edges of it are colored the same. An edge-coloring*graph*G is*rainbow*connected if any two vertices are connected by a*rainbow**path*. ...##
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Ramsey numbers in rainbow triangle free colorings

2010
*
The Australasian Journal of Combinatorics
*

Given a

dblp:journals/ajc/FaudreeGJM10
fatcat:rf5yu4f2v5ghroitxbr45wztcu
*graph*G, we consider the problem of finding the minimum number n such that any k edge colored*complete**graph*on n vertices contains either a three colored triangle or a monochromatic copy of the ...*graph*G. ... Proof: Consider a*complete**graph*on n − 1 vertices colored entirely*with*color 1. This*graph*certainly contains*no**rainbow*triangle or monochromatic copy of G. ...##
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Note on rainbow connection number of dense graphs
[article]

2011
*
arXiv
*
pre-print

An edge-colored

arXiv:1110.1268v1
fatcat:xjp2ciabtfhu5dasl26kbrhelm
*graph*G is*rainbow*connected if any two vertices are connected by a*path*whose edges have distinct colors. ... We show that for k≥ 2, if G is a non-*complete**graph*of order n*with*minimum degree δ (G)≥n/2-1+log_kn, or minimum degree-sum σ_2(G)≥ n-2+2log_kn, then rc(G)≤ k; if G is a*graph*of order n*with*diameter ... If there is*no**path*connecting u and v, we set d(x, y) := ∞. An edge-coloring of a*graph*is a function from its edges set to the set of natural numbers. ...##
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Rainbow Connectivity: Hardness and Tractability

2011
*
Foundations of Software Technology and Theoretical Computer Science
*

A

doi:10.4230/lipics.fsttcs.2011.241
dblp:conf/fsttcs/AnanthNS11
fatcat:m5qbhvfzd5b2xhqernmuadoce4
*path*in an edge colored*graph*is said to be a*rainbow**path*if*no*two edges on the*path*have the same color. ... An edge colored*graph*is (strongly)*rainbow*connected if there exists a (geodesic)*rainbow**path*between every pair of vertices. ... A*path*between a pair of vertices is said to be a*rainbow**path*, if*no*two edges on the*path*have the same color. ...##
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Complexity of rainbow vertex connectivity problems for restricted graph classes

2017
*
Discrete Applied Mathematics
*

The

doi:10.1016/j.dam.2016.11.023
fatcat:h5ud5mtswzfizeg5jvdbd6yupy
*graph*G is said to be*rainbow*vertex connected if there is a vertex*rainbow**path*between every pair of its vertices. ... A*path*in a vertex-colored*graph*G is vertex*rainbow*if all of its internal vertices have a distinct color. ... However, as each vertex in {f j | 1 ≤ j < m} ∪ {t } is a cut vertex colored*with*color c j ,*no*vertex*rainbow**path*R from s 0 to t can use vertex h j . ...##
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Hardness results for total rainbow connection of graphs

2016
*
Discussiones Mathematicae Graph Theory
*

We show that deciding whether a given total-coloring of a

doi:10.7151/dmgt.1856
fatcat:l52vqw4oubejvk62fofswwcfvy
*graph*G makes it total*rainbow*connected is NP-*Complete*. We also prove that given a*graph*G, deciding whether trc(G) = 3 is NP-*Complete*. ... is, any two vertices of G are connected by a total*rainbow**path*. ... Let G be a given*graph**with*an edge-coloring c. ...##
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Hardness result for the total rainbow k-connection of graphs
[article]

2015
*
arXiv
*
pre-print

A

arXiv:1511.06119v1
fatcat:pllkaujbungwfbpnieowwq4js4
*path*in a total-colored*graph*is called total*rainbow*if its edges and internal vertices have distinct colors. ... In this paper, we study the computational complexity of total*rainbow*k-connection number of*graphs*. We show that it is NP-*complete*to decide whether trc_k(G)=3. ... A*path*in G is called a*rainbow**path*if*no*two edges of the*path*are colored the same. The*graph*G is called*rainbow*connected if for any two vertices of G, there is a*rainbow**path*connecting them. ...##
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Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs

2018
*
International Symposium on Mathematical Foundations of Computer Science
*

Given a

doi:10.4230/lipics.mfcs.2018.83
dblp:conf/mfcs/HeggernesILLL18
fatcat:3zrivhwysbgxrliwami56jyynu
*graph**with*colors on its vertices, a*path*is called a*rainbow*vertex*path*if all its internal vertices have distinct colors. ... We say that the*graph*is*rainbow*vertex-connected if there is a*rainbow*vertex*path*between every pair of its vertices. ... A*rainbow**path*in G is a*path*all of whose edges are colored*with*distinct colors, and G is*rainbow*-connected if there is a*rainbow**path*between every pair of its vertices. ...##
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The rainbow 2-connectivity of Cartesian products of 2-connected graphs and paths

2020
*
Electronic Journal of Graph Theory and Applications
*

We also determine the

doi:10.5614/ejgta.2020.8.1.11
doaj:b295703a0ad74c12ba940c338e4f45e1
fatcat:5g27eifuijalxfviego6qvfj5y
*rainbow*2-connection number of the Cartesian products of some*graphs*, i.e.*complete**graphs*, fans, wheels, and cycles,*with**paths*. ... An edge-colored*graph*G is*rainbow*k-connected, if there are k-internally disjoint*rainbow**paths*connecting every pair of vertices of G. ... An edge-colored*path*P in G is*rainbow*if*no*two edges of P are colored the same. ...
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