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Complete bipartite graphs with no rainbow paths
2016
International Journal of Contemporary Mathematical Sciences
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In this paper, we consider the edge colorings of 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. ...
doi:10.12988/ijcms.2016.6951
fatcat:7layqmcvrjc7pmmirleo3guk24
Note on the complexity of deciding the rainbow connectedness for bipartite graphs
[article]
2011
arXiv
pre-print
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A 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. ...
arXiv:1109.5534v2
fatcat:lkfy6t6zb5hedbwrrq2ezvrny4
Complexity results for rainbow matchings
2014
Theoretical Computer Science
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The latter is somewhat surprising, since, to the best of our knowledge, 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. ...
doi:10.1016/j.tcs.2013.12.013
fatcat:inezqohlejgedn433llc72fuvm
Rainbow connections for planar graphs and line graphs
[article]
2011
arXiv
pre-print
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We also give upper bounds of the 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 ...
arXiv:1110.3147v2
fatcat:fsizn4hurndh7gg5myhjf7gvcy
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
[article]
2018
arXiv
pre-print
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A 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. ...
arXiv:1405.6893v7
fatcat:gobg6k74yreybbn5s3hruy6gbm
Further hardness results on rainbow and strong rainbow connectivity
2016
Discrete Applied Mathematics
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2014 pre-print arXiv:1404.3082v3 fatcat:2oucuajidjasjf7gj45kfbv6jmOther Versions
A 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. ...
doi:10.1016/j.dam.2015.07.041
fatcat:brax3buo7jbwzgfvwf2luyrqqy
The rainbow connection of windmill and corona graph
2014
Applied Mathematical Sciences
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In this paper we determine the exact values of the windmill 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. ...
doi:10.12988/ams.2014.48632
fatcat:to6swntqxvgknabze6jpdhhlyy
Ramsey numbers in rainbow triangle free colorings
2010
The Australasian Journal of Combinatorics
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Given a 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. ...
dblp:journals/ajc/FaudreeGJM10
fatcat:rf5yu4f2v5ghroitxbr45wztcu
Note on rainbow connection number of dense graphs
[article]
2011
arXiv
pre-print
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An edge-colored 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. ...
arXiv:1110.1268v1
fatcat:xjp2ciabtfhu5dasl26kbrhelm
Rainbow Connectivity: Hardness and Tractability
2011
Foundations of Software Technology and Theoretical Computer Science
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A 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. ...
doi:10.4230/lipics.fsttcs.2011.241
dblp:conf/fsttcs/AnanthNS11
fatcat:m5qbhvfzd5b2xhqernmuadoce4
Complexity of rainbow vertex connectivity problems for restricted graph classes
2017
Discrete Applied Mathematics
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2016 pre-print arXiv:1612.07768v1 fatcat:5gd4ud5mkfd6vgi6fphh63kynm
The 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 . ...
doi:10.1016/j.dam.2016.11.023
fatcat:h5ud5mtswzfizeg5jvdbd6yupy
Hardness results for total rainbow connection of graphs
2016
Discussiones Mathematicae Graph Theory
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We show that deciding whether a given total-coloring of a 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. ...
doi:10.7151/dmgt.1856
fatcat:l52vqw4oubejvk62fofswwcfvy
Hardness result for the total rainbow k-connection of graphs
[article]
2015
arXiv
pre-print
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A 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. ...
arXiv:1511.06119v1
fatcat:pllkaujbungwfbpnieowwq4js4
Rainbow Vertex Coloring Bipartite Graphs and Chordal Graphs
2018
International Symposium on Mathematical Foundations of Computer Science
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Given a 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. ...
doi:10.4230/lipics.mfcs.2018.83
dblp:conf/mfcs/HeggernesILLL18
fatcat:3zrivhwysbgxrliwami56jyynu
The rainbow 2-connectivity of Cartesian products of 2-connected graphs and paths
2020
Electronic Journal of Graph Theory and Applications
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We also determine the 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. ...
doi:10.5614/ejgta.2020.8.1.11
doaj:b295703a0ad74c12ba940c338e4f45e1
fatcat:5g27eifuijalxfviego6qvfj5y
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