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Complete bipartite graphs with no rainbow paths

Kaijun Wang, Huaping Wang, Lifen Li
2016 International Journal of Contemporary Mathematical Sciences  
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]

Shasha Li, Xueliang Li
2011 arXiv   pre-print
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

Van Bang Le, Florian Pfender
2014 Theoretical Computer Science  
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]

Xiaolong Huang, Xueliang Li, Yongtang Shi
2011 arXiv   pre-print
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]

Melissa Keranen, Juho Lauri
2018 arXiv   pre-print
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

Juho Lauri
2016 Discrete Applied Mathematics  
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

Yixiao Liu, Zhiping Wang
2014 Applied Mathematical Sciences  
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

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson, Colton Magnant
2010 The Australasian Journal of Combinatorics  
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]

Jiuying Dong, Xueliang Li
2011 arXiv   pre-print
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

Prabhanjan Ananth, Meghana Nasre, Kanthi K. Sarpatwar, Marc Herbstritt
2011 Foundations of Software Technology and Theoretical Computer Science  
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

Juho Lauri
2017 Discrete Applied Mathematics  
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

Lily Chen, Bofeng Huo, Yingbin Ma
2016 Discussiones Mathematicae Graph Theory  
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]

Wenjing Li, Xueliang Li, Di Wu
2015 arXiv   pre-print
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

Pinar Heggernes, Davis Issac, Juho Lauri, Paloma T. Lima, Erik Jan Van Leeuwen, Michael Wagner
2018 International Symposium on Mathematical Foundations of Computer Science  
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

Bety Hayat Susanti, A.N.M. Salman, Rinovia Simanjuntak
2020 Electronic Journal of Graph Theory and Applications  
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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