Maximal r-Matching Sequences of Graphs and...
Transcript of Maximal r-Matching Sequences of Graphs and...
![Page 1: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/1.jpg)
Maximal r -Matching Sequences of Graphs andHypergraphs
Adam Mammoliti
26th February 2018
![Page 2: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/2.jpg)
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.
K6
![Page 3: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/3.jpg)
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.
K6
![Page 4: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/4.jpg)
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.
K6
![Page 5: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/5.jpg)
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.
K6
![Page 6: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/6.jpg)
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.
K6
![Page 7: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/7.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 8: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/8.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 9: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/9.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 10: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/10.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 11: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/11.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 12: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/12.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 13: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/13.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 14: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/14.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 15: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/15.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
![Page 16: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/16.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
cms(C7) = 3
![Page 17: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/17.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
cms(C7) = 3
![Page 18: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/18.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
cms(C7) = 3
![Page 19: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/19.jpg)
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(`)}
C7
1
2
3
4
5
67
ms(C7) = 3
cms(C7) = 3
![Page 20: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/20.jpg)
Known Results
Alspach (2008)
ms(Kn) = ⌊n − 1
2⌋
Brualdi et al. (2012)
cms(Kn) = ⌊n − 2
2⌋
![Page 21: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/21.jpg)
Known Results
Alspach (2008)
ms(Kn) = ⌊n − 1
2⌋
Brualdi et al. (2012)
cms(Kn) = ⌊n − 2
2⌋
![Page 22: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/22.jpg)
Known Results
Brualdi et al. (2012)
cms(Cn) = ms(Cn) = ⌊n − 1
2⌋
Brualdi et al. (2012)
ms(Pn) = ⌊n − 1
2⌋ and cms(Pn) = ⌊n − 2
2⌋
![Page 23: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/23.jpg)
Known Results
Brualdi et al. (2012)
cms(Cn) = ms(Cn) = ⌊n − 1
2⌋
Brualdi et al. (2012)
ms(Pn) = ⌊n − 1
2⌋ and cms(Pn) = ⌊n − 2
2⌋
![Page 24: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/24.jpg)
Known Results
Kreher et al. (2015)
cms(G) ≥ ⌊ms(G)2
⌋
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⌊n−12
⌋ edges ⌈n−12
⌉ edges
![Page 25: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/25.jpg)
Known Results
Kreher et al. (2015)
cms(G) ≥ ⌈ms(G)2
⌉
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⌊n−12
⌋ edges ⌈n−12
⌉ edges
![Page 26: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/26.jpg)
Known Results
Kreher et al. (2015)
cms(G) ≥ ⌈ms(G)2
⌉
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⌊n−12
⌋ edges ⌈n−12
⌉ edges
![Page 27: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/27.jpg)
Known Results
Kreher et al. (2015)
cms(G) ≥ ⌈ms(G)2
⌉
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⌊n−12
⌋ edges ⌈n−12
⌉ edges
![Page 28: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/28.jpg)
Known Results
Kreher et al. (2015)
cms(G) ≥ ⌈ms(G)2
⌉
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅
n − 1 edges
⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⌊n−12
⌋ edges ⌈n−12
⌉ edges
![Page 29: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/29.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
deg(v) = 5
K6
v
![Page 30: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/30.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
deg(v) = 5
K6
v
![Page 31: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/31.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
deg(v) = 5
K6
v
![Page 32: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/32.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
deg(v) = 5
K6
v
![Page 33: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/33.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ V
msr(G) = max`
{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 34: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/34.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 35: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/35.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 36: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/36.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 37: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/37.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 38: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/38.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 39: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/39.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 40: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/40.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 41: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/41.jpg)
Generalising Matching Sequencibility
deg(v) = number of edges touching v
msr(`) = analogy of ms(`) with deg(v) ≤ r for all v ∈ Vmsr(G) = max
`{msr(`)}
K4
1
3
2
4
5 6
r = 2
ms2(K4) = 3
![Page 42: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/42.jpg)
Generalising Cyclic Matching Sequencibility
deg(v) = number of edges touching v
cmsr(`) = analogy of msr(`) over cyclically consecutive edges
cmsr(G) = max`
{cmsr(`)}
K4
r = 2
ms2(K4) = 31
3
2
4
5 6
cms2(K4) = 3
![Page 43: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/43.jpg)
Generalising Cyclic Matching Sequencibility
deg(v) = number of edges touching v
cmsr(`) = analogy of msr(`) over cyclically consecutive edges
cmsr(G) = max`
{cmsr(`)}
K4
r = 2
ms2(K4) = 31
3
2
4
5 6
cms2(K4) = 3
![Page 44: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/44.jpg)
Generalising Cyclic Matching Sequencibility
deg(v) = number of edges touching v
cmsr(`) = analogy of msr(`) over cyclically consecutive edges
cmsr(G) = max`
{cmsr(`)}
K4
r = 2
ms2(K4) = 31
3
2
4
5 6
cms2(K4) = 3
![Page 45: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/45.jpg)
Generalising Cyclic Matching Sequencibility
deg(v) = number of edges touching v
cmsr(`) = analogy of msr(`) over cyclically consecutive edges
cmsr(G) = max`
{cmsr(`)}
K4
r = 2
ms2(K4) = 31
3
2
4
5 6cms2(K4) = 3
![Page 46: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/46.jpg)
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
2⌋
and if n is even or r = n−12 , then
cmsr(Kn) = ⌊ rn − 1
2⌋
Theorem
If r is even and n is odd, then
⌊ rn − 1
2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1
2⌋
![Page 47: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/47.jpg)
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
2⌋
and if n is even or r = n−12 , then
cmsr(Kn) = ⌊ rn − 1
2⌋
Theorem
If r is even and n is odd, then
⌊ rn − 1
2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1
2⌋
![Page 48: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/48.jpg)
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
2⌋
and if n is even or r = n−12 , then
cmsr(Kn) = ⌊ rn − 1
2⌋
Theorem
If r is even and n is odd, then
⌊ rn − 1
2⌋ − 1 ≤ cmsr(Kn) ≤ ⌊ rn − 1
2⌋
![Page 49: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/49.jpg)
Remaining cases?
Conjecture
msr(Kn) = ⌊ rn − 1
2⌋
Conjecture
If rn is even, then
cmsr(Kn) = ⌊ rn − 1
2⌋
![Page 50: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/50.jpg)
Remaining cases?
Conjecture
msr(Kn) = ⌊ rn − 1
2⌋
Conjecture
If rn is even, then
cmsr(Kn) = ⌊ rn − 1
2⌋
![Page 51: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/51.jpg)
Remaining cases?
Question
For what odd r and n does
cmsr(Kn) = ⌊ rn − 1
2⌋
Theorem
If r and n are odd, then
cmsr(Kn) = ⌊ rn − 1
2⌋ iff cmsn−1−r(Kn) = ⌊(n − 1 − r)n − 1
2⌋
![Page 52: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/52.jpg)
Remaining cases?
Question
For what odd r and n does
cmsr(Kn) = ⌊ rn − 1
2⌋
Theorem
If r and n are odd, then
cmsr(Kn) = ⌊ rn − 1
2⌋ iff cmsn−1−r(Kn) = ⌊(n − 1 − r)n − 1
2⌋
![Page 53: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/53.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 54: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/54.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 55: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/55.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 56: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/56.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 57: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/57.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 58: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/58.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 59: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/59.jpg)
Sketch of the Construction: r = 1
cms(K8) = 3
K8
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1−1
2−2
3−3
∞
0
1
2
3
4
1−1
2−2
3−3
∞
0
5 6
78
![Page 60: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/60.jpg)
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 , . . .
![Page 61: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/61.jpg)
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 , . . .
![Page 62: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/62.jpg)
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 , . . .
![Page 63: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/63.jpg)
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 , . . .
![Page 64: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/64.jpg)
Sketch of the Construction: General r
cms3(K16) = 3×162 − 1 = 23
K16
v1,0
v2,0
v−1,0
v−2,0
v1,1
v2,1
v−1,1
v−2,1
v1,−1
v2,−1
v−1,−1
v−2,−1
v0,0
v0,1
v0,−1
v∞3
6
4
7
2
5
-3
-6
-2
-5
-4
-7
0
1
−1
∞
![Page 65: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/65.jpg)
Sketch of the Construction: General r
cms3(K16) = 3×162 − 1 = 23
K16
v1,0
v2,0
v−1,0
v−2,0
v1,1
v2,1
v−1,1
v−2,1
v1,−1
v2,−1
v−1,−1
v−2,−1
v0,0
v0,1
v0,−1
v∞
3
6
4
7
2
5
-3
-6
-2
-5
-4
-7
0
1
−1
∞
![Page 66: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/66.jpg)
Sketch of the Construction: General r
cms3(K16) = 3×162 − 1 = 23
K16
v1,0
v2,0
v−1,0
v−2,0
v1,1
v2,1
v−1,1
v−2,1
v1,−1
v2,−1
v−1,−1
v−2,−1
v0,0
v0,1
v0,−1
v∞
3
6
4
7
2
5
-3
-6
-2
-5
-4
-7
0
1
−1
∞
![Page 67: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/67.jpg)
Sketch of the Construction: General r
cms3(K16) = 3×162 − 1 = 23
K16
v1,0
v2,0
v−1,0
v−2,0
v1,1
v2,1
v−1,1
v−2,1
v1,−1
v2,−1
v−1,−1
v−2,−1
v0,0
v0,1
v0,−1
v∞3
6
4
7
2
5
-3
-6
-2
-5
-4
-7
0
1
−1
∞
![Page 68: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/68.jpg)
Sketch of the Construction: General r
cms3(K16) = 3×162 − 1 = 23
K16
v1,0
v2,0
v−1,0
v−2,0
v1,1
v2,1
v−1,1
v−2,1
v1,−1
v2,−1
v−1,−1
v−2,−1
v0,0
v0,1
v0,−1
v∞
3
6
4
7
2
5
-3
-6
-2
-5
-4
-7
0
1
−1
∞
![Page 69: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/69.jpg)
Complete Bipartite Graphs
K2,3
![Page 70: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/70.jpg)
Complete Bipartite Graphs
Brualdi et al. (2012)
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
![Page 71: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/71.jpg)
Complete Bipartite Graphs
Brualdi et al. (2012)
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
![Page 72: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/72.jpg)
Further Generalisations
Theorem
If n ≤ m, then
msr(Kn,m) = cmsr(Kn,m) =⎧⎪⎪⎨⎪⎪⎩
rn if n < m
rn − 1 if n = m
Theorem
If n1 < n2 ≤ ⋯ ≤ nk , then
msr(Kn1,n2,...,nk ) = cmsr(Kn1,n2,...,nk ) = rn1
![Page 73: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/73.jpg)
Further Generalisations
Theorem
Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk .
Then
msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ r or (1) below, holds
rn1 − 1 otherwise
and
cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ rrn1 − 1 otherwise
where
(⌊ r
nu−11
⌋ + 1)⌊1r
k
∏i=2
ni⌋ ≤k
∏i=u+1
ni ≤ ⌊ r
nu−11
⌋(⌊1r
k
∏i=2
ni⌋ + 1) (1)
![Page 74: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/74.jpg)
Further Generalisations
Theorem
Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk . Then
msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ r or (1) below, holds
rn1 − 1 otherwise
and
cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ rrn1 − 1 otherwise
where
(⌊ r
nu−11
⌋ + 1)⌊1r
k
∏i=2
ni⌋ ≤k
∏i=u+1
ni ≤ ⌊ r
nu−11
⌋(⌊1r
k
∏i=2
ni⌋ + 1) (1)
![Page 75: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/75.jpg)
Further Generalisations
Theorem
Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk . Then
msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ r or (1) below, holds
rn1 − 1 otherwise
and
cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ rrn1 − 1 otherwise
where
(⌊ r
nu−11
⌋ + 1)⌊1r
k
∏i=2
ni⌋ ≤k
∏i=u+1
ni ≤ ⌊ r
nu−11
⌋(⌊1r
k
∏i=2
ni⌋ + 1) (1)
![Page 76: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/76.jpg)
Further Generalisations
Theorem
Let 1 ≤ n1 = n2 = ⋯ = nu < nu+1 ≤ ⋯ ≤ nk . Then
msr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ r or (1) below, holds
rn1 − 1 otherwise
and
cmsr(Kn1,...,nk ) =⎧⎪⎪⎨⎪⎪⎩
rn1 if nu−11 ∣ rrn1 − 1 otherwise
where
(⌊ r
nu−11
⌋ + 1)⌊1r
k
∏i=2
ni⌋ ≤k
∏i=u+1
ni ≤ ⌊ r
nu−11
⌋(⌊1r
k
∏i=2
ni⌋ + 1) (1)
![Page 77: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/77.jpg)
The End
Thanks for listening!
![Page 78: Maximal r-Matching Sequences of Graphs and Hypergraphsusers.monash.edu/...r...of-Graphs-and-Hypergraphs.pdf · Maximal r-Matching Sequences of Graphs and Hypergraphs Adam Mammoliti](https://reader035.fdocuments.us/reader035/viewer/2022062507/5fc5a15dbc1a1842c43de1d4/html5/thumbnails/78.jpg)
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.