maximal r-matching sequences of graphs and...

Maximal r -Matching Sequences of Graphs andHypergraphs

Adam Mammoliti

26th February 2018

Definitions and Examples

Kn is the graph with n vertices and every possible edge.

A matching is a set of disjoint edges in a graph.

Matching Sequencibility

A labelling is a bijective function ` ∶ E → {1,2, . . . , ∣E ∣}.ms(`) = max{s ∶ every s consecutive edges are a matching}ms(G) = max

`{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

A labelling is a bijective function ` ∶ E → {1,2, . . . , ∣E ∣}.

ms(`) = max{s ∶ every s consecutive edges are a matching}ms(G) = max

`{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

A labelling is a bijective function ` ∶ E → {1,2, . . . , ∣E ∣}.ms(`) = max{s ∶ every s consecutive edges are a matching}

ms(G) = max`

{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

`{ms(`)}

ms(C7) = 3

Cyclic Matching Sequencibility

A labelling is a bijective function ` ∶ E → {1,2, . . . , ∣E ∣}.cms(`) = analogy of ms(`) over cyclically consecutive edges

cms(G) = max`

{cms(`)}

ms(C7) = 3

cms(C7) = 3

cms(G) = max`

{cms(`)}

ms(C7) = 3

cms(C7) = 3

cms(G) = max`

{cms(`)}

ms(C7) = 3

cms(C7) = 3

cms(G) = max`

{cms(`)}

ms(C7) = 3

cms(C7) = 3

Known Results

Alspach (2008)

ms(Kn) = ⌊n − 1

Brualdi et al. (2012)

cms(Kn) = ⌊n − 2

Known Results

Alspach (2008)

ms(Kn) = ⌊n − 1

cms(Kn) = ⌊n − 2

Known Results

cms(Cn) = ms(Cn) = ⌊n − 1

ms(Pn) = ⌊n − 1

2⌋ and cms(Pn) = ⌊n − 2

Known Results

cms(Cn) = ms(Cn) = ⌊n − 1

ms(Pn) = ⌊n − 1

2⌋ and cms(Pn) = ⌊n − 2

Known Results

Kreher et al. (2015)

cms(G) ≥ ⌊ms(G)2

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⌊n−12

⌋ edges ⌈n−12

⌉ edges

Known Results

cms(G) ≥ ⌈ms(G)2

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⌊n−12

⌋ edges ⌈n−12

⌉ edges

Known Results

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⌊n−12

⌋ edges ⌈n−12

⌉ edges

Known Results

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⌊n−12

⌋ edges ⌈n−12

⌉ edges

Known Results

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅

n − 1 edges

⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⌊n−12

⌋ edges ⌈n−12

⌉ edges

Generalising Matching Sequencibility

deg(v) = number of edges touching v

deg(v) = 5

msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ V

msr(G) = max`

{msr(`)}

ms2(K4) = 3

msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

`{msr(`)}

ms2(K4) = 3

Generalising Cyclic Matching Sequencibility

cmsr(`) = analogy of msr(`) over cyclically consecutive edges

cmsr(G) = max`

{cmsr(`)}

ms2(K4) = 31

cms2(K4) = 3

cmsr(G) = max`

{cmsr(`)}

ms2(K4) = 31

cms2(K4) = 3

cmsr(G) = max`

{cmsr(`)}

ms2(K4) = 31

cms2(K4) = 3

cmsr(G) = max`

{cmsr(`)}

ms2(K4) = 31

5 6cms2(K4) = 3

Generalisations

Theorem

If rn is even or n is odd and either r ≥ n−12 or gcd(r ,n − 1) = 1, then

msr(Kn) = ⌊ rn − 1

and if n is even or r = n−12 , then

cmsr(Kn) = ⌊ rn − 1

Theorem

If r is even and n is odd, then

⌊ rn − 1

2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1

Generalisations

Theorem

msr(Kn) = ⌊ rn − 1

Theorem

⌊ rn − 1

2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1

Generalisations

Theorem

msr(Kn) = ⌊ rn − 1

Theorem

⌊ rn − 1

2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1

Remaining cases?

Conjecture

msr(Kn) = ⌊ rn − 1

Conjecture

If rn is even, then

Remaining cases?

Conjecture

msr(Kn) = ⌊ rn − 1

Conjecture

If rn is even, then

Remaining cases?

Question

For what odd r and n does

Theorem

If r and n are odd, then

2⌋ iff cmsn−1−r(Kn) = ⌊(n − 1 − r)n − 1

Remaining cases?

Question

For what odd r and n does

Theorem

If r and n are odd, then

2⌋ iff cmsn−1−r(Kn) = ⌊(n − 1 − r)n − 1

Sketch of the Construction: r = 1

cms(K8) = 3

Sketch of the Construction: General r

cmsr(Kn) = rn2 − 1

Start with a matching decomposition M1M2⋯Mn−1 of Kn

A subsequence of rn2 − 1 edges is of the form

e1⋯ ej´¹¹¹¹¹¸¹¹¹¹¹¹¶

edges in Mi

Mi+1⋯Mi+r−1 ej+1⋯ en−1´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶edges in Mi+r

The matchings r spaces apart form the collections

M1,Mr+1, . . . M2,Mr+2, . . . ⋯ Md ,Mr+d , . . .

e1⋯ ej´¹¹¹¹¹¸¹¹¹¹¹¹¶

edges in Mi

M1,Mr+1, . . . M2,Mr+2, . . . ⋯ Md ,Mr+d , . . .

e1⋯ ej´¹¹¹¹¹¸¹¹¹¹¹¹¶

edges in Mi

M1,Mr+1, . . . M2,Mr+2, . . . ⋯ Md ,Mr+d , . . .

e1⋯ ej´¹¹¹¹¹¸¹¹¹¹¹¹¶

edges in Mi

M1,Mr+1, . . . M2,Mr+2, . . . ⋯ Md ,Mr+d , . . .

cms3(K16) = 3×162 − 1 = 23

v−1,0

v−2,0

v−1,1

v−2,1

v1,−1

v2,−1

v−1,−1

v−2,−1

v0,−1

cms3(K16) = 3×162 − 1 = 23

v−1,0

v−2,0

v−1,1

v−2,1

v1,−1

v2,−1

v−1,−1

v−2,−1

v0,−1

cms3(K16) = 3×162 − 1 = 23

v−1,0

v−2,0

v−1,1

v−2,1

v1,−1

v2,−1

v−1,−1

v−2,−1

v0,−1

cms3(K16) = 3×162 − 1 = 23

v−1,0

v−2,0

v−1,1

v−2,1

v1,−1

v2,−1

v−1,−1

v−2,−1

v0,−1

cms3(K16) = 3×162 − 1 = 23

v−1,0

v−2,0

v−1,1

v−2,1

v1,−1

v2,−1

v−1,−1

v−2,−1

v0,−1

Complete Bipartite Graphs

If n ≤ m, then

ms(Kn,m) = cms(Kn,m) =⎧⎪⎪⎨⎪⎪⎩

n if n < m

n − 1 if n = m

Theorem

If n ≤ m, then

msr(Kn,m) = cmsr(Kn,m) =⎧⎪⎪⎨⎪⎪⎩

rn if n < m

rn − 1 if n = m

Complete Bipartite Graphs

If n ≤ m, then

ms(Kn,m) = cms(Kn,m) =⎧⎪⎪⎨⎪⎪⎩

n if n < m

n − 1 if n = m

Theorem

If n ≤ m, then

rn if n < m

rn − 1 if n = m

Further Generalisations

Theorem

If n ≤ m, then

rn if n < m

rn − 1 if n = m

Theorem

If n1 < n2 ≤ ⋯ ≤ nk , then

msr(Kn1,n2,...,nk ) = cmsr(Kn1,n2,...,nk ) = rn1

Theorem

Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk .

msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

rn1 if nu−11 ∣ r or (1) below, holds

rn1 − 1 otherwise

cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

rn1 if nu−11 ∣ rrn1 − 1 otherwise

(⌊ r

nu−11

⌋ + 1)⌊1r

∏i=2

ni⌋ ≤k

∏i=u+1

ni ≤ ⌊ r

nu−11

⌋(⌊1r

∏i=2

ni⌋ + 1) (1)

Theorem

Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk . Then

msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

rn1 − 1 otherwise

cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

(⌊ r

nu−11

⌋ + 1)⌊1r

∏i=2

ni⌋ ≤k

∏i=u+1

ni ≤ ⌊ r

nu−11

⌋(⌊1r

∏i=2

ni⌋ + 1) (1)

Theorem

msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

rn1 − 1 otherwise

cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

(⌊ r

nu−11

⌋ + 1)⌊1r

∏i=2

ni⌋ ≤k

∏i=u+1

ni ≤ ⌊ r

nu−11

⌋(⌊1r

∏i=2

ni⌋ + 1) (1)

Theorem

msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

rn1 − 1 otherwise

cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩

(⌊ r

nu−11

⌋ + 1)⌊1r

∏i=2

ni⌋ ≤k

∏i=u+1

ni ≤ ⌊ r

nu−11

⌋(⌊1r

∏i=2

ni⌋ + 1) (1)

The End

Thanks for listening!

References

B. Alspach. The wonderful Walecki construction.Bull. Inst. Combin. Appl. 52 (2008), 7–12.

R. A. Brualdi, K. P. Kiernan, S. A. Meyer, M. W. Schroeder.Cyclic matching sequencibility of graphs.Australas. J. Combin. 53 (2012), 245–256.

D. L. Kreher, A. Pastine, L. Tollefson.A note on the cyclic sequencibility of graphs.Australas. J. Combin. 61(2) (2015), 142–146.

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