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Upper Bounding Rainbow Connection Number by Forest Number [article]

L. Sunil Chandran, Davis Issac, Juho Lauri, Erik Jan van Leeuwen
2020 arXiv   pre-print
The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G).  ...  In this work we show that if we consider the forest number f(G), the number of vertices in a maximum induced forest of G, instead of t(G), then surprisingly we do get an upper bound.  ...  The minimum number of colors required to rainbow-connect G is known as the rainbow connection number of G, and denoted as rc(G).  ... 
arXiv:2006.06551v1 fatcat:e2hjhupqsfhxhfozqnj76ygowu

Some upper bounds for 3-rainbow index of graphs [article]

Tingting Liu, Yumei Hu
2013 arXiv   pre-print
The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rx_k(G). In this paper, we consider 3-rainbow index rx_3(G) of G.  ...  We first show that for connected graph G with minimum degree δ(G)≥ 3, the tight upper bound of rx_3(G) is rx_3(G[D])+4, where D is the connected 2-dominating set of G.  ...  Upper bounds for 3-rainbow index of general graphs In this section, we derive a sharp bound for 3-rainbow index of general graphs by block decomposition.  ... 
arXiv:1310.2355v1 fatcat:zn2ddnmii5fqpercef6kwvsvdy

Upper Bounds for 4-rainbow Index of Graphs

Shumin ZHANG
2018 DEStech Transactions on Engineering and Technology Research  
And then we determine a tight upper bound for Ks,t (4 ≤ s ≤ t) and a better bound for (P5,C5)-free graphs. Finally, we obtain a sharp bound for 4-rainbow index of general graphs.  ...  In this paper, we study 4-rainbow index rx4(G) of G.  ...  Next, we determine the upper bound for 4-rainbow index of (P 5 ,C 5 )-free graphs.  ... 
doi:10.12783/dtetr/apop2017/18707 fatcat:jywyqllnqzezpdxnn62ecii5ua

Rainbow connection number and connected dominating sets

L. Sunil Chandran, Anita Das, Deepak Rajendraprasad, Nithin M. Varma
2011 Journal of Graph Theory  
In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γ_c(G) + 2, where γ_c(G) is the connected domination number of  ...  An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) +  ...  Hence order of the graph minus one is an upper bound for rainbow connection number.  ... 
doi:10.1002/jgt.20643 fatcat:p3y2c25y65ge5marvxv6b3wfli

Upper bounds for the rainbow connection numbers of line graphs [article]

Xueliang Li, Yuefang Sun
2010 arXiv   pre-print
The rainbow connection number of G, denoted by rc(G), is defined as the smallest number of colors by using which there is a coloring such that G is rainbow connected.  ...  In this paper, we mainly study the rainbow connection number of the line graph of a graph which contains triangles and get two sharp upper bounds for rc(L(G)), in terms of the number of edge-disjoint triangles  ...  For example, in [3] the authors got a very good upper bound for the rainbow connection number of a 2-connected graph according to their ear-decomposition.  ... 
arXiv:1001.0287v1 fatcat:qkg2pmoo5ve3vabz35i3qhz44y

Rainbow Connection Number and Radius [article]

Manu Basavaraju, L. Sunil Chandran, Deepak Rajendraprasad, and Arunselvan Ramaswamy
2012 arXiv   pre-print
The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no  ...  We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity.  ...  Such upper bounds were shown for some special graph classes in [3] . But, for a general graph, the rainbow connection number cannot be upper bounded by a function of r alone.  ... 
arXiv:1011.0620v2 fatcat:hi7effujrnabboi6e7fuljwuk4

Rainbow connections of graphs -- A survey [article]

Xueliang Li, Yuefang Sun
2011 arXiv   pre-print
We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number  ...  The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it.  ...  They gave two sharp upper bounds for rainbow connection number of a line graph and one sharp upper bound for rainbow connection number of an iterated line graph.  ... 
arXiv:1101.5747v2 fatcat:zwwaavvzzjbwtjvn4wujnauibu

Anti-Ramsey Numbers for Graphs with Independent Cycles

Zemin Jin, Xueliang Li
2009 Electronic Journal of Combinatorics  
The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$.  ...  An edge-colored graph is called rainbow if all the colors on its edges are distinct. Let ${\cal G}$ be a family of graphs.  ...  Anti-Ramsey numbers were introduced by Erdős et al. in [5] , and showed to be connected not so much to Ramsey theory than to Turán numbers.  ... 
doi:10.37236/174 fatcat:lznvgkslsraa7hykaf47gpulhu

Good upper bounds for the total rainbow connection of graphs [article]

Hui Jiang, Xueliang Li, Yingying Zhang
2015 arXiv   pre-print
The total rainbow connection number of G, denoted by trc(G), is defined as the smallest number of colors that are needed to make G total-rainbow connected.  ...  A total-colored graph G is total-rainbow connected if any two vertices of G are connected by a total rainbow path of G.  ...  The problem of determining the number rvc(G) of a connected graph G is also NP-hard; see [7, 8] . There are a few results about the upper bounds of the rainbow vertex-connection number.  ... 
arXiv:1501.01806v1 fatcat:up7ygcfm7jbjrdilinhpnefoyu

Note on the upper bound of the rainbow index of a graph [article]

Qingqiong Cai, Xueliang Li, Yan Zhao
2014 arXiv   pre-print
The rainbow connection number of a connected graph G, denoted by rc(G), is the minimum number of colors that are needed to color the edges of G such that there exists a rainbow path connecting every two  ...  The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow tree connecting S for each k-subset S of V(G) is called the k-rainbow index of G, denoted by rx_k(G),  ...  The minimum number of colors for which there is an edge-coloring of G such that G is rainbow connected is called the rainbow connection number, denoted by rc(G).  ... 
arXiv:1407.4663v2 fatcat:ri745ysp4vcslkpafkubwxeoji

Page 7620 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
That is, for a connected upper bound graph G, G* is an upper bound graph if and only if for any pair of A*-simplicial vertices 5), 5» such that dg(s,, 52) <k, there exists a G* -simplicial vertex § satis  ...  Bounds on the bipartite rainbow Ramsey number for various classes of pairs of graphs are given, and some precise numbers are determined.  ... 

Rainbow connection of graphs with diameter 2 [article]

Hengzhe Li, Xueliang Li, Sujuan Liu
2011 arXiv   pre-print
The rainbow connection number rc(G) of G is the minimum integer i for which there exists an i-edge-coloring of G such that every two distinct vertices of G are connected by a rainbow path.  ...  So, it is interesting to know the best upper bound of rc(G) for such a graph G.  ...  However, we failed to find an example for which the rainbow connection number reaches 5.  ... 
arXiv:1101.2765v2 fatcat:o5pt3abpkvb6xivkppetc2oduq

Rainbow connection of graphs with diameter 2

Hengzhe Li, Xueliang Li, Sujuan Liu
2012 Discrete Mathematics  
The rainbow connection number rc(G) of G is the minimum integer k for which there exists a k-edge-coloring of G such that any two distinct vertices of G are connected by a rainbow path.  ...  So, it is interesting to know the upper bound of rc(G) for such a graph G.  ...  In this paper, we give sharp upper bounds on the rainbow connection number of a graph with diameter 2: if G is a bridgeless graph with diameter 2, then rc(G) ≤ 5; if G is a connected graph with diameter  ... 
doi:10.1016/j.disc.2012.01.009 fatcat:h7allx2f5jdvhj4vyugygprquq

Upper bound for the rainbow connection number of bridgeless graphs with diameter 3 [article]

Hengzhe Li, Xueliang Li, Yuefang Sun
2011 arXiv   pre-print
The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path.  ...  In this paper, we prove that a bridgeless graph with diameter 3 has rainbow connection number at most 9.  ...  Bounds for the rainbow connection number of a graph have also been studies in terms of other graph parameters, for example, radius, diameter, dominating number, minimum degree, connectivity, etc.  ... 
arXiv:1109.2769v2 fatcat:w3wk7hoyabcztpwhbvr3ocawga

On degree anti-Ramsey numbers

Shoni Gilboa, Dan Hefetz
2017 European journal of combinatorics (Print)  
In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey  ...  The degree anti-Ramsey number AR_d(H) of a graph H is the smallest integer k for which there exists a graph G with maximum degree at most k such that any proper edge colouring of G yields a rainbow copy  ...  Note that the upper bounds on AR d (C k ) we proved in Theorem 1.4 entail upper bounds on the size anti-Ramsey numbers of cycles.  ... 
doi:10.1016/j.ejc.2016.09.002 fatcat:6e4cnp5cr5gfzlbvukseoh356a
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