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Upper Bounding Rainbow Connection Number by Forest Number
[article]

2020
*
arXiv
*
pre-print

The minimum

arXiv:2006.06551v1
fatcat:e2hjhupqsfhxhfozqnj76ygowu
*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). ...##
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Some upper bounds for 3-rainbow index of graphs
[article]

2013
*
arXiv
*
pre-print

The minimum

arXiv:1310.2355v1
fatcat:zn2ddnmii5fqpercef6kwvsvdy
*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. ...##
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Upper Bounds for 4-rainbow Index of Graphs

2018
*
DEStech Transactions on Engineering and Technology Research
*

And then we determine a tight

doi:10.12783/dtetr/apop2017/18707
fatcat:jywyqllnqzezpdxnn62ecii5ua
*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. ...##
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Rainbow connection number and connected dominating sets

2011
*
Journal of Graph Theory
*

In this paper we show that for every

doi:10.1002/jgt.20643
fatcat:p3y2c25y65ge5marvxv6b3wfli
*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*. ...##
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Upper bounds for the rainbow connection numbers of line graphs
[article]

2010
*
arXiv
*
pre-print

The

arXiv:1001.0287v1
fatcat:qkg2pmoo5ve3vabz35i3qhz44y
*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. ...##
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Rainbow Connection Number and Radius
[article]

2012
*
arXiv
*
pre-print

The

arXiv:1011.0620v2
fatcat:hi7effujrnabboi6e7fuljwuk4
*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. ...##
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Rainbow connections of graphs -- A survey
[article]

2011
*
arXiv
*
pre-print

We begin with an introduction, and then try to organize the work into five categories, including (strong)

arXiv:1101.5747v2
fatcat:zwwaavvzzjbwtjvn4wujnauibu
*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. ...##
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Anti-Ramsey Numbers for Graphs with Independent Cycles

2009
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Electronic Journal of Combinatorics
*

The anti-Ramsey

doi:10.37236/174
fatcat:lznvgkslsraa7hykaf47gpulhu
*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*. ...##
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Good upper bounds for the total rainbow connection of graphs
[article]

2015
*
arXiv
*
pre-print

The total

arXiv:1501.01806v1
fatcat:up7ygcfm7jbjrdilinhpnefoyu
*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*. ...##
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Note on the upper bound of the rainbow index of a graph
[article]

2014
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arXiv
*
pre-print

The

arXiv:1407.4663v2
fatcat:ri745ysp4vcslkpafkubwxeoji
*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). ...##
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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. ...##
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Rainbow connection of graphs with diameter 2
[article]

2011
*
arXiv
*
pre-print

The

arXiv:1101.2765v2
fatcat:o5pt3abpkvb6xivkppetc2oduq
*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. ...##
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Rainbow connection of graphs with diameter 2

2012
*
Discrete Mathematics
*

The

doi:10.1016/j.disc.2012.01.009
fatcat:h7allx2f5jdvhj4vyugygprquq
*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 ...##
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Upper bound for the rainbow connection number of bridgeless graphs with diameter 3
[article]

2011
*
arXiv
*
pre-print

The

arXiv:1109.2769v2
fatcat:w3wk7hoyabcztpwhbvr3ocawga
*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. ...##
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On degree anti-Ramsey numbers

2017
*
European journal of combinatorics (Print)
*

In this paper we prove a general

doi:10.1016/j.ejc.2016.09.002
fatcat:6e4cnp5cr5gfzlbvukseoh356a
*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. ...
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