Rainbow Connection in Hypergraphs - cantab.net rc hypergraphs slides.pdf · Rainbow Connection in...

Rainbow Connection in Hypergraphs

Henry Liu

Universidade Nova de Lisboa, Portugal

Joint work with Rui Carpentier, Manuel Silva and Teresa Sousa

Bordeaux Graph Workshop, 21 to 24 November 2012

IntroductionHypergraph Cycles

Complete Multipartite HypergraphsSeparation Results

Introduction

All graphs and hypergraphs are simple and finite.

DefinitionThe rainbow connection number rc(G ) of a connected graph G isthe minimum number of colours needed to colour the edges of Gsuch that, any two vertices are connected by a path with distinctcolours (i.e., a rainbow path).

The function rc(G ) was first introduced by Chartrand, Johns,McKeon and Zhang in 2008.

The study of rc(G ) has since attracted a lot of interest. Manygeneralisations and variant functions have been considered.Recently, a survey by Li, Shi and Sun, and a book by Li and Sun,were published on the rainbow connection subject.

Henry Liu Rainbow Connection in Hypergraphs



Example (Chartrand et al., 2008)

rc(G ) = e(G ) if and only if G is a tree.

Example (Chartrand et al., 2008)

rc(C3) = 1, and rc(Cn) = dn2e if n ≥ 4.

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C6

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C7




We consider the rainbow connection notion for hypergraphs.

But, what is a path in a hypergraph?

DefinitionA (Berge) path consists of distinct vertices v1, . . . , v`+1 anddistinct edges e1, . . . , e` such that vi , vi+1 ∈ ei for 1 ≤ i ≤ `.

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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3

DefinitionThe rainbow connection number rc(H) of a connected hypergraphH is the minimum number of colours needed to colour the edges ofH such that, any two vertices are connected by a rainbow path.




We consider the rainbow connection notion for hypergraphs.But, what is a path in a hypergraph?


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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3







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v1 v2e1

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3







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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2

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e3







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v1 v2e1 . . . ...................................................................................................................................................................

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v3e2 . .............................................................................................................................................................................................................................................................................................

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v4

e3





DefinitionFor 1 ≤ s < r , an (r , s)-path is an r-uniform interval hypergraphwhere every two consecutive edges intersect in s vertices.

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(5, 2)-path of length 3

DefinitionThe (r , s)-rainbow connection number rc(H, r , s) of a connectedhypergraph H is the minimum number of colours needed to colourthe edges of H such that, any two vertices are connected by arainbow (r , s)-path (if the minimum exists).

Hence, we have two generalisations of rc(G ) to hypergraphs.




DefinitionA connected hypergraph H is minimally connected if H− e isdisconnected for all e ∈ E (H).

Theorem 1 (CLSS, 2012)

Let H be a connected hypergraph with e(H) ≥ 1. Then,rc(H) = e(H) if and only if H is minimally connected.

Remark

It is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

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rc(H) = 2e(H) = 3

Minimally connected hypergraphs and hypertrees are two ratherdifferent families, unlike in the graphs setting.







RemarkIt is not true that rc(H) = e(H) if and only if H is a hypertree.For example, let H = · · ·

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rc(H) = 2e(H) = 3









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rc(H) = 2

e(H) = 3









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rc(H) = 2e(H) = 3





Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

..

(7, 3)-cycle

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Hypergraph Cycles

DefinitionFor n > r ≥ 2, the (n, r)-cycle is the r -uniform hypergraph Cr

n ...

.....

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(7, 3)-cycle

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rc(Crn, r , s) well-defined for all 1 ≤ s < r . We consider r ≥ 3.





Let n > r ≥ 3. Then for sufficiently large n,

(a) rc(Crn) = rc(Cr

n, r , 1) = d n2(r−1)e.

(b) rc(Crn, r , r − 1) = dn2e.

(c) rc(Crn, r , s) ∈ d , d + 1 for 1 ≤ s ≤ r − 2, where

d = dn+1−2s2(r−s) e, the “ (r , s)-diameter of Cr

n”.

Hence, rc(Crn, r , s) = d n

2(r−s)e+ O(1) for all 1 ≤ s < r .

Proof (sketch).

Lower bound: Computation of d is easy ⇒ (a) (except for n ≡ 1(mod 2(r − 1))) and (c). Remaining lower bounds also easy.

Upper bound: Colour a “wraparound” (r , s)-path with the requirednumber of colours.





Let n > r ≥ 3. Then for sufficiently large n,

(a) rc(Crn) = rc(Cr

n, r , 1) = d n2(r−1)e.

(b) rc(Crn, r , r − 1) = dn2e.

(c) rc(Crn, r , s) ∈ d , d + 1 for 1 ≤ s ≤ r − 2, where

d = dn+1−2s2(r−s) e, the “ (r , s)-diameter of Cr

n”.

Hence, rc(Crn, r , s) = d n

2(r−s)e+ O(1) for all 1 ≤ s < r .

Proof (sketch).

Lower bound: Computation of d is easy ⇒ (a) (except for n ≡ 1(mod 2(r − 1))) and (c). Remaining lower bounds also easy.Upper bound: Colour a “wraparound” (r , s)-path with the requirednumber of colours.




Complete Multipartite Hypergraphs

DefinitionFor t ≥ r ≥ 2 and 1 ≤ n1 ≤ · · · ≤ nt , the r -uniform hypergraphKr

n1,...,ntis ...

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K32,2,3,4

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Krn1,...,nt

is a complete multipartite hypergraph.




rc(K2n1,...,nt

) determined by Chartrand et al. Let r ≥ 3.


Let t ≥ r ≥ 3 and 1 ≤ n1 ≤ · · · ≤ nt . Then,

rc(Krn1,...,nt

) =

1 if nt = 1,

2 if nt−1 ≥ 2, or t > r , nt−1 = 1 and nt ≥ 2,

nt if t = r and nt−1 = 1.

Proof not too difficult. For the last case, Krn1,...,nt

is minimallyconnected.More interesting to consider rc(Kr

n1,...,nt, r , s).




rc(K2n1,...,nt




rc(Krn1,...,nt

) =

1 if nt = 1,




is minimallyconnected.

More interesting to consider rc(Krn1,...,nt

, r , s).




rc(K2n1,...,nt




rc(Krn1,...,nt

) =

1 if nt = 1,




is minimallyconnected.More interesting to consider rc(Kr

n1,...,nt, r , s).





Let t ≥ r ≥ 3, s ≤ r − 2 and 1 ≤ n1 ≤ · · · ≤ nt be such thatnt = 1 or n2(t−r)+s+1 ≥ 2. Then,

rc(Krn1,...,nt

, r , s) =

1 if nt = 1,

2 if nt ≥ 2.

Proof still not difficult. But surprisingly, rc(Krn1,...,nt

, r , r − 1) is alot more difficult to determine ...





Let t ≥ r ≥ 3, 1 ≤ n1 ≤ · · · ≤ nt , n = nt and b =∑

S

∏i∈S ni ,

where S ranges over all subsets of 1, . . . , t − 1 with size r − 1.Then,

rc(Krn1,...,nt

, r , r − 1) =

1 if nt = 1,

d b√

n e if t = r and n1 = 1,

min(d b√

n e, r + 2) if t = r and n1 ≥ 2,

min(d b√

n e, 3) if t > r .




Proof (sketch).

Upper bound rc(Krn1,...,nt

, r , r − 1) ≤ d b√

n e =: k .

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n1 n2 · · · nt−1

nt.w .................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. w 6= w ′ get distinct tuples

If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

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c

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.⇒

i1 i2 ir−1

This is a good k-colouring.rc(Kr

n1,...,nt, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; and

rc(Krn1,...,nt

, r , r − 1) ≤ 3 if t > r : similar idea.




Proof (sketch).


, r , r − 1) ≤ d b√

n e =: k .

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n1 n2 · · · nt−1

nt

.w .................................(t−1r−1


If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

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c

. . .

.⇒

i1 i2 ir−1



rc(Krn1,...,nt





Proof (sketch).


, r , r − 1) ≤ d b√

n e =: k .

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n1 n2 · · · nt−1

nt.w .................................(t−1r−1


If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

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c

. . .

.⇒

i1 i2 ir−1



rc(Krn1,...,nt





Proof (sketch).


, r , r − 1) ≤ d b√

n e =: k .

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n1 n2 · · · nt−1

nt.w .................................(t−1r−1


If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

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.............

c

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.⇒

i1 i2 ir−1

This is a good k-colouring.

rc(Krn1,...,nt

, r , r − 1) ≤ r + 2 if t = r and n1 ≥ 2; andrc(Kr

n1,...,nt, r , r − 1) ≤ 3 if t > r : similar idea.




Proof (sketch).


, r , r − 1) ≤ d b√

n e =: k .

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n1 n2 · · · nt−1

nt.w .................................(t−1r−1


If W Sj (i1, . . . , ir−1) = c ∈ 1, . . . , k

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c

. . .

.⇒

i1 i2 ir−1



rc(Krn1,...,nt





Proof (sketch, ctd.).

Lower bound for t > r : rc(Krn1,...,nt

, r , r − 1) ≥ min(d b√

n e, 3).

Takea 2-colouring.

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n1 n2 · · · nt−1

nt

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.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1

)-tuple of functions W S1 , W S2 , . . . s.t. W Sj (i1, . . . , ir−1) = c

Then ∃w , w ′ with the same(t−1r−1

)-tuples ⇒ @ rainbow w − w ′

(r , r − 1)-path, and the 2-colouring is bad.Other lower bounds: similar.






, r , r − 1) ≥ min(d b√

n e, 3). Takea 2-colouring.

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n1 n2 · · · nt−1

nt

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.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1










, r , r − 1) ≥ min(d b√


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n1 n2 · · · nt−1

nt

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.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1



)-tuples

⇒ @ rainbow w − w ′







, r , r − 1) ≥ min(d b√


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n1 n2 · · · nt−1

nt

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.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1




(r , r − 1)-path, and the 2-colouring is bad.

Other lower bounds: similar.






, r , r − 1) ≥ min(d b√


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n1 n2 · · · nt−1

nt

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.............

c ∈ 1, 2

.w

. . .

.

i1 i2 ir−1

.................................(t−1r−1








Separation Results


Let a > 0, r ≥ 3 and 1 ≤ s 6= s ′ < r . Then, there exists anr-uniform hypergraph H such that rc(H, r , s)− rc(H) ≥ a, andthere exists an r-uniform hypergraph H such thatrc(H, r , s)− rc(H, r , s ′) ≥ a.

Proof.First part with s > 1, and second part with s > s ′, take H = Cr

n.⇒ Difference is at least

n

2(r − s)− n

2(r − s ′)+ O(1) ≥ Ω(n).




Proof (ctd.).

First part with s = 1 and second part with s < s ′, take thefollowing construction for H.

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. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

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Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.




Proof (ctd.).


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. .s

r

u v

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r − s ′ − 1,“fixed”

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Proof (ctd.).


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. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

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Proof (ctd.).


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. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

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Above colouring ⇒ rc(H) = rc(H, r , s ′) = 2.

Considering u and v ⇒ rc(H, r , s) ≥ length of (r , s)-path.




Proof (ctd.).


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. .s

r

u v...................................................................................... ....... .................................................................. .......

s ′r − s ′ − 1,

“fixed”

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References

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

2. X. Li, Y. Shi, and Y. Sun, Rainbow connections of graphs - asurvey, Graphs Combin., to appear.

3. X. Li, and Y. Sun, “Rainbow Connections of Graphs”,Springer-Verlag, New York, 2012, viii+103pp.




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