Some Remarks on Rainbow Connectivity

Kamčev, Nina; Krivelevich, Michael; Sudakov, Benny

doi:10.1002/jgt.22003

Cited by 16 publications

(9 citation statements)

References 22 publications

(60 reference statements)

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“…As mentioned earlier, rc(G) ≤ n − 1 follows by coloring the edges of a spanning tree of G with distinct colors. Moreover, Kamčev et al [10] proved that rc(G) ≤ diam(G 1 ) + diam(G 2 ) + c, where G 1 = (V, E 1 ) and G 2 = (V, E 2 ) are connected spanning subgraphs of G and c ≤ |E 1 ∩ E 2 |. For a more comprehensive treatment, we refer the curious reader to the books [6,14] and the surveys [13,15] on rainbow connectivity.…”

Section: Introductionmentioning

confidence: 99%

Upper Bounding Rainbow Connection Number by Forest Number

Chandran¹,

Issac²,

Lauri³

et al. 2020

Preprint

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A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and the graph is rainbow-connected if there is a rainbow path between each pair of its vertices. The minimum number of colors needed to rainbow-connect a graph G is the rainbow connection number of G, denoted by rc(G).A simple way to rainbow-connect a graph G is to color the edges of a spanning tree with distinct colors and then re-use any of these colors to color the remaining edges of G. This proves that rc(G) ≤ |V (G)| − 1. We ask whether there is a stronger connection between tree-like structures and rainbow coloring than that is implied by the above trivial argument. For instance, is it possible to find an upper bound of t(G) − 1 for rc(G), where t(G) is the number of vertices in the largest induced tree of G? The answer turns out to be negative, as there are counter-examples that show that even c • t(G) is not an upper bound for rc(G) for any given constant c.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. More specifically, we prove that rc(G) ≤ f(G) + 2. Our result indicates a stronger connection between rainbow connection and tree-like structures than that was suggested by the simple spanning tree based upper bound.

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Section: Introductionmentioning

confidence: 99%

Upper Bounding Rainbow Connection Number by Forest Number

Chandran¹,

Issac²,

Lauri³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Chakraborty et al [5] showed that given a graph G, deciding if rc(G) = 2 is NP-complete. Bounds for the rainbow connection number of a graph have also been studied in terms of other graph parameters, see [14,17,19,18,22] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Conflict-free connection number and independence number of a graph

Wang

2020

Preprint

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An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cf c(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cf c(G) ≤ α(G) for any connected graph G, and an example is given showing that the bound is sharp. With this result, we prove that if T is a tree with ∆(T ) ≥ α(T )+2 2 , then cf c(T ) = ∆(T ).

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“…Frieze and Tsourakakis [12] studied the rainbow connection of a random graph at the connectivity threshold. Dudek, Frieze and Tsourakakis [13] and Kamcev, Krivelevich and Sudakov [16] and Molloy [18] studied the rainbow connection of random regular graphs. Suffice it to say that in general the rainbow connection is close to the diameter in all cases.…”

Section: Introductionmentioning

confidence: 99%

Pattern Colored Hamilton Cycles in Random Graphs

Anastos¹,

Frieze²

2019

SIAM J. Discrete Math.

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We consider the existence of patterned Hamilton cycles in randomly colored random graphs. Given a string Π over a set of colors {1, 2, . . . , r}, we say that a Hamilton cycle is Π-colored if the pattern repeats at intervals of length |Π| as we go around the cycle. We prove a hitting time for the existence of such a cycle. We also prove a hitting time result for a related notion of Π-connected.The next phase of this study, concerns Hamilton cycles in random k-uniform hypergraphs. There are various notions of Hamilton cycle in this context, and Ferber and Krivelevich [7] proved that if m H edges are needed for a given type of hamilton cycle to exist w.h.p. then O(m H ) random edges and (1 + ǫ)m H colors are sufficient for a rainbow Hamilton cycles. The hypergraph results in [7] were sharpened by Dudek, English and Frieze [5]. Cooper and Frieze [4] considered the related question of what is the threshold for every k-bounded coloring of the edges of G n,m to contain at least one rainbow Hamilton cycle.Here k-bounded means that no color can be used more than k times.

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Some Remarks on Rainbow Connectivity

Cited by 16 publications

References 22 publications

Upper Bounding Rainbow Connection Number by Forest Number

Upper Bounding Rainbow Connection Number by Forest Number

Conflict-free connection number and independence number of a graph

Pattern Colored Hamilton Cycles in Random Graphs

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