Rainbow k-connection in Dense Graphs - Alumni Liu Magnant ec11 slides.pdf · Introduction Graphs...

Rainbow k-connection in Dense Graphs

Shinya Fujita1, Henry Liu∗2, Colton Magnant3

1Gunma National College of Technology, Japan2Universidade Nova de Lisboa, Portugal3Georgia Southern University, GA, USA

EuroComb’11, Budapest, August/September 2011

IntroductionGraphs with Fixed Connectivity

Complete Bipartite and Multipartite GraphsRandom GraphsOpen Problems

Introduction

I G is a finite, simple, k-connected graph (k ∈ N).

I An edge-coloured path is rainbow if its edges have distinctcolours.

I An edge-colouring (not necessarily proper) for G is rainbowk-connected if any two vertices of G are connected by kinternally vertex-disjoint rainbow paths.

I The rainbow k-connection number of G , denoted by rck(G ),is the minimum integer s such that there exists a rainbowk-connected edge-colouring of G , using s colours.

I Write rc(G ) = rc1(G ).

I Note: rck(G ) is well-defined if G is k-connected (by Menger’sTheorem).

Shinya Fujita, Henry Liu∗, Colton Magnant Rainbow k-connection in Dense Graphs

Introduction

Example

G = C5 + v , the wheel with five spokes.

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Example

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⇒ rc(G ) = 2.

Example

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⇒ rc(G ) = 2.

Example

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⇒ rc2(G ) = 3.

Example

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⇒ rc2(G ) = 3.

Example

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⇒ rc3(G ) = 5.

Example

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⇒ rc3(G ) = 5.

I Introduced by Chartrand, Johns, McKeon, and Zhang in 2008.

I They studied rc(G ) and rck(G ) for some basic graphs,including complete and complete bipartite graphs.

I They introduced a related function: the strong rainbowconnection number src(G ).

I They presented an interesting application to secure datatransfer.

I Since then, the function rck(G ) has been studied by manypeople. Many results about rck(G ) have been proved when Gsatisfies some condition, such as in relation to minimumdegree, diameter, connectivity, ... of G . Further relatedfunctions to rck(G ) have been introduced.

I Survey paper recently written by Li and Sun (2011).

Graphs with Fixed Connectivity

Question 1 (Broersma, 2009)

If G is an `-connected graph, then what is rc(G )?

Theorem 2If G is an `-connected graph on n vertices, then

I ` = 2: rc(G ) ≤ 2n3 and rc(G ) ≤ n

2 + O(√

n) (Caro, Lev,Roditty, Tuza, and Yuster, 2008).

I ` = 3: rc(G ) ≤ 3(n+1)5 (Li and Shi, 2010).

I General `: rc(G ) ≤ 3nδ(G)+1 + 3. ⇒ rc(G ) ≤ 3n

`+1 + 3

(Chandran, Das, Rajendraprasad, and Varma, 2010).

I ` = 2: rc(G ) ≤ 2n3 and rc(G ) ≤ n

2 + O(√

I ` = 3: rc(G ) ≤ 3(n+1)5 (Li and Shi, 2010).

`+1 + 3

I ` = 2: rc(G ) ≤ 2n3 and rc(G ) ≤ n

2 + O(√

I ` = 3: rc(G ) ≤ 3(n+1)5 (Li and Shi, 2010).

`+1 + 3

I ` = 2: rc(G ) ≤ 2n3 and rc(G ) ≤ n

2 + O(√

I ` = 3: rc(G ) ≤ 3(n+1)5 (Li and Shi, 2010).

`+1 + 3

I ` = 2: rc(G ) ≤ 2n3 and rc(G ) ≤ n

2 + O(√

I ` = 3: rc(G ) ≤ 3(n+1)5 (Li and Shi, 2010).

`+1 + 3

Question 3 (Li and Sun, 2010)

If ` ≥ 2 and G is an `-connected graph, then what is rc2(G )?

Theorem 4 (Fujita, L., Magnant, 2011)

If ` ≥ 2 and G is an `-connected graph with on n vertices, thenrc2(G ) ≤ (`+1)n

Proof (sketch).

First, construct a spanning subgraph H ⊂ G as follows.

I Take a cycle H0 ⊂ G with |V (H0)| ≥ `.

I Repeatedly attach subdivided K1,`’s (by Menger’s Theorem),until all vertices of G are exhausted. Let the graphs obtainedbe H0 ⊂ H1 ⊂ H2 ⊂ · · · ⊂ Ht = H, with V (H) = V (G ).

Proof (sketch).

Example

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H3, etc...

I It suffices to find a rainbow 2-connected colouring for Hi usingat most (`+1)|V (Hi )|

` colours, for every 0 ≤ i ≤ t.

I In fact, we prove that the colouring for Hi satisfies thefollowing.

For all u, v ∈ V (Hi ), there are two internally vertex-disjointrainbow u − v paths.

For all u ∈ V (Hi ) and X ⊂ V (Hi ) with |X | = 2, there are twodisjoint rainbow u − X paths (except at u).

For all X , Y ⊂ V (Hi ) with |X | = |Y | = 2, there are twodisjoint rainbow X − Y paths.

I Use induction on i . For i = 0, take a rainbow colouring of H0.Then, |V (H0)| < (`+1)|V (H0)|

` colours are used, and (a) to (c)are satisfied.

` colours, for every 0 ≤ i ≤ t.I In fact, we prove that the colouring for Hi satisfies the

following.

For all u, v ∈ V (Hi ), there are two internally vertex-disjointrainbow u − v paths.

following.

(a) For all u, v ∈ V (Hi ), there are two internally vertex-disjointrainbow u − v paths.

following.

(b) For all u ∈ V (Hi ) and X ⊂ V (Hi ) with |X | = 2, there are twodisjoint rainbow u − X paths (except at u).

following.

(c) For all X , Y ⊂ V (Hi ) with |X | = |Y | = 2, there are twodisjoint rainbow X − Y paths.

following.

(c) For all X , Y ⊂ V (Hi ) with |X | = |Y | = 2, there are twodisjoint rainbow X − Y paths.

I For 1 ≤ i ≤ t, assume that the whole claim holds for Hi−1.

One possible case is shown. For this case, colour Hi as follows.

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Hi−1

I For 1 ≤ i ≤ t, assume that the whole claim holds for Hi−1.One possible case is shown.

For this case, colour Hi as follows.

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Hi−1

I For 1 ≤ i ≤ t, assume that the whole claim holds for Hi−1.One possible case is shown. For this case, colour Hi as follows.

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Hi−1

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Hi−1

Colour the other new edges with new, distinct colours.

I Using a case by case analysis, it is easy to check that at most(`+1)|V (Hi )|

` colours are used, and (a) to (c) hold for Hi . �

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Hi−1

Colour the other new edges with new, distinct colours.

I Using a case by case analysis, it is easy to check that at most(`+1)|V (Hi )|

` colours are used, and (a) to (c) hold for Hi . �

If G is a 2-connected, series-parallel graph on n vertices, thenrc2(G ) ≤ n.

Roughly speaking, a 2-connected, series-parallel graph is a graphwhich can be obtained from a cycle, followed by repeatedlyattaching ‘ears’ in a specific way. These graphs are a rather largesubfamily of the 2-connected graphs.The proof of Theorem 5 is similar to that of Theorem 4. But toconsider the colouring for G , we need to embed G into the planeand turn it into a directed graph.

Roughly speaking, a 2-connected, series-parallel graph is a graphwhich can be obtained from a cycle, followed by repeatedlyattaching ‘ears’ in a specific way. These graphs are a rather largesubfamily of the 2-connected graphs.

The proof of Theorem 5 is similar to that of Theorem 4. But toconsider the colouring for G , we need to embed G into the planeand turn it into a directed graph.

Roughly speaking, a 2-connected, series-parallel graph is a graphwhich can be obtained from a cycle, followed by repeatedlyattaching ‘ears’ in a specific way. These graphs are a rather largesubfamily of the 2-connected graphs.The proof of Theorem 5 is similar to that of Theorem 4. But toconsider the colouring for G , we need to embed G into the planeand turn it into a directed graph.

Complete Bipartite and Multipartite Graphs

Theorem 6 (Chartrand et al, 2008)

If t ≥ 2, 1 ≤ n1 ≤ · · · ≤ nt ,∑t−1

i=1 ni = m and nt = n, then

rc(Kn1,...,nt ) =

n if t = 2 and n1 = 1,

min(d m√

n e, 4) if t = 2 and 2 ≤ n1 ≤ n2,

1 if t ≥ 3 and nt = 1,

2 if t ≥ 3, nt ≥ 2 and m > n,

min(d m√

n e, 3) if t ≥ 3 and m ≤ n.

Theorem 7 (Li and Sun, 2011)

If k ≥ 2 and n ≥ 2kdk2 e, then rck(Kn,n) = 3.

The proof of Theorem 7 considers an explicit 3-colouring of Kn,n.

Let ε > 0. There exists a function f (ε) such that if k ≥ f (ε) andn ≥ (2 + ε)k, then rck(Kn,n) = 3.

Proof (sketch).

I Colour a perfect matching of Kn,n with colour 1. Randomlycolour the other edges with colours 2 and 3.

I Let Euv be the event that there are no k internallyvertex-disjoint rainbow u − v paths. Using the Chernoff andunion bounds, P(

⋃Euv ) < 1 for sufficiently large n. �

Proof (sketch).

Let Kt×n be the complete t-partite graph, with each class having nvertices.

Let t ≥ 3 and ε > 0. There exists a function g(ε) such that ifk ≥ g(ε) and n ≥ ( 2

t−2 + ε)k, then rck(Kt×n) = 2.

The proof of Theorem 9 is similar to that of Theorem 8. Here wejust consider a random 2-colouring of Kt×n.

Random Graphs

DefinitionLet Q be a graph property of Gn,p, where p = p(n). A functionf (n) is a sharp threshold function for Q if there are constantsc, C > 0 such that, Gn,cf (n) does not satisfy Q a.s., and Gn,p

satisfies Q a.s. for all p ≥ Cf (n).

A result of Bollobas and Thomason (1986) implies that theproperty rck(Gn,p) ≤ d (for some d ≥ 2) has a threshold function.

Random Graphs

DefinitionLet Q be a graph property of Gn,p, where p = p(n). A functionf (n) is a sharp threshold function for Q if there are constantsc, C > 0 such that, Gn,cf (n) does not satisfy Q a.s., and Gn,p

satisfies Q a.s. for all p ≥ Cf (n).

A result of Bollobas and Thomason (1986) implies that theproperty rck(Gn,p) ≤ d (for some d ≥ 2) has a threshold function.

For all k ≥ 1, p =√

log n/n is a sharp threshold function for theproperty rck(Gn,p) ≤ 2.

The case k = 1 was proved by Caro et al (2008). A generalisationof this to the property rck(Gn,p) ≤ d (for some d ≥ 2), withk ≤ O(log n), was proved by He and Liang (2010).

log n/n is a sharp threshold function for theproperty rck(Gn,p) ≤ 2.

The case k = 1 was proved by Caro et al (2008). A generalisationof this to the property rck(Gn,p) ≤ d (for some d ≥ 2), withk ≤ O(log n), was proved by He and Liang (2010).

Proof of Theorem 10 (sketch).

I Firstly, we prove that for some C > 0 and p ≥ C√

log n/n,any two vertices of Gn,p have at least 4 log2 n commonneighbours, a.s. This then allows us to use a standardprobabilistic argument, involving a random 2-colouring, andthe union bound, to show that rck(Gn,p) = 2.

I Secondly, we prove that for some c > 0 and p = c√

log n/n,Gn,p has diameter at least 3, a.s. To do this, we prove thatthere is a set A ⊂ V (Gn,p) such that a.s., A is an independentset, and a.s., there are two vertices of A with no commonneighbour outside A. �

log n/n is a sharp threshold function for theproperty rck(Gn,n,p) ≤ 3.(Gn,n,p is the random bipartite graph with two classes of size n,and edge probability p).

For all k ≥ 1, M =√

n3 log n is a sharp threshold function for theproperty rck(Gn,M) ≤ 2.

The proofs of Theorems 11 and 12 are similar to that of Theorem10.

Open Problems

For graphs with fixed connectivity, we can ask the followingquestion.

Problem 13 (Fujita, L., Magnant, 2011)

Let 1 ≤ k ≤ `. Find the least constant c = c(k , `), where0 < c ≤ k, such that for all `-connected graphs G on n vertices,we have rck(G ) ≤ cn.

The bound c ≤ k follows from a result of Mader (1972).

Open Problems

For complete bipartite and multipartite graphs, we can ask thefollowing question.

For k, t ≥ 2 and n1 ≤ · · · ≤ nt , is there a function h(k , t) suchthat, if n1 ≥ h(k , t), then

rck(Kn1,...,nt ) =

{3 if t = 2,

2 if t ≥ 3?

The case t = 2 (bipartite case) was asked by Chartrand et al(2009).

rck(Kn1,...,nt ) =

{3 if t = 2,

2 if t ≥ 3?

rck(Kn1,...,nt ) =

{3 if t = 2,

2 if t ≥ 3?

Concerning random graphs, we can ask the following question.

Find a threshold function for another random graph model.

In particular, an answer for random regular graphs will beinteresting.

References

1. Y. Caro, A. Lev, Y. Roditty, Zs. Tuza, and R. Yuster, On rainbowconnection, Electron. J. Combin. 15(1) (2008), #R57, 13pp.

2. G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, Rainbowconnection in graphs, Math. Bohem. 133(1) (2008), 85-98.

3. G. Chartrand, G. L. Johns, K. A. McKeon, and P. Zhang, Therainbow connectivity of a graph, Networks 54(2) (2009), 75-81.

4. J. He, and H. Liang, On rainbow-k-connectivity of random graphs,arXiv:1012.1942v1 (2010).

5. M. Krivelevich, and R. Yuster, The rainbow connection of a graph is(at most) reciprocal to its minimum degree, J. Graph Th. 63(3)(2009), 185-191.

6. X. Li, and Y. Sun, Rainbow connections of graphs - a survey,arXiv:1101.5747v2 (2011).

Thank you!

Rainbow k-connection in Dense Graphs - Alumni Liu Magnant ec11 slides.pdf · Introduction Graphs...

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