Average Connectivity and Average Edge-connectivity in Graphs Suil O - - PowerPoint PPT Presentation

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Average Edge-connectivity in Graphs

Suil O joint work with Jaehoon Kim

University of Illinois at Urbana-Champaign

24th Cumberland Conference on Combinatorics, Graph Theory, and Computing (May 12th 2011)

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Table of Contents

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Connectivity and Edge-connectivity

◮ The connectivity of G, written κ(G), is the minimum size of a

vertex set S such that G − S is disconnected.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Connectivity and Edge-connectivity

◮ The connectivity of G, written κ(G), is the minimum size of a

vertex set S such that G − S is disconnected.

◮ The edge-connectivity of G, written κ′(G), is the minimum

size of an edge set F such that G − F is disconnected.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Connectivity and Edge-connectivity

◮ The connectivity of G, written κ(G), is the minimum size of a

vertex set S such that G − S is disconnected.

◮ The edge-connectivity of G, written κ′(G), is the minimum

size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Connectivity and Edge-connectivity

◮ The connectivity of G, written κ(G), is the minimum size of a

vertex set S such that G − S is disconnected.

◮ The edge-connectivity of G, written κ′(G), is the minimum

size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart. However, since these values are based on a worst-case situation, it does not reflect the “global (edge) connectedness” of the graph.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Connectivity and Edge-connectivity

◮ The connectivity of G, written κ(G), is the minimum size of a

vertex set S such that G − S is disconnected.

◮ The edge-connectivity of G, written κ′(G), is the minimum

size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart. However, since these values are based on a worst-case situation, it does not reflect the “global (edge) connectedness” of the graph.

Figure: TwoGraphsG1 andG2 with connectivity 1 and edge-connectivity 1

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Average Edge-connectivity

The average connectivity of a graph G with n vertices, written κ(G), is

P

u,v∈V (G) κ(u,v)

(n

2)

, where κ(u, v) is the minimum number of vertices whose deletion makes v unreachable from u.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Average Edge-connectivity

The average connectivity of a graph G with n vertices, written κ(G), is

P

u,v∈V (G) κ(u,v)

(n

2)

, where κ(u, v) is the minimum number of vertices whose deletion makes v unreachable from u. The average edge-connectivity of a graph G with n vertices, written κ′(G), is

P

u,v∈V (G) κ′(u,v)

(n

2)

, where κ′(u, v) is the minimum number of edges whose deletion makes v unreachable from u.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Average Edge-connectivity

The average connectivity of a graph G with n vertices, written κ(G), is

P

u,v∈V (G) κ(u,v)

(n

2)

, where κ(u, v) is the minimum number of vertices whose deletion makes v unreachable from u. The average edge-connectivity of a graph G with n vertices, written κ′(G), is

P

u,v∈V (G) κ′(u,v)

(n

2)

, where κ′(u, v) is the minimum number of edges whose deletion makes v unreachable from u.

Figure: κ(G1) = κ′(G1) = 27

7 and κ(G2) = κ′(G2) = 12 7

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Matching Number

In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Matching Number

In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ(G) ≤ d.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Matching Number

In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ(G) ≤ d.

We prove a bound on the average connectivity in terms of matching number.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Matching Number

In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ(G) ≤ d.

We prove a bound on the average connectivity in terms of matching number. Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G), and this is sharp.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity and Matching Number

In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ(G) ≤ d.

We prove a bound on the average connectivity in terms of matching number. Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G), and this is sharp. Furthermore, if G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G), and this is sharp.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, k(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, k(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Proof:

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, k(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Proof:

◮ Let M be a maximum matching in G

and let S = V (G) − M.

M S=V(G)−M

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, k(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Proof:

◮ Let M be a maximum matching in G

and let S = V (G) − M.

◮ For vv′ ∈ M, put v and v′ into T, T ′ and R as

follows: If neither v nor v′ has a neighbor in S, then put both in T. If v′ has a neighbor in S and v does not, then put v in T and v′ in T ′.

M S=V(G)−M T T T’ R R

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, k(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Proof:

◮ Let M be a maximum matching in G

and let S = V (G) − M.

◮ For vv′ ∈ M, put v and v′ into T, T ′ and R as

follows: If neither v nor v′ has a neighbor in S, then put both in T. If v′ has a neighbor in S and v does not, then put v in T and v′ in T ′.

◮ If both have neighbors in S, put them both in R. M S=V(G)−M R R

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Consider three cases to obtain upper bounds on κ(u, v) depending

• n the possible locations of distinct vertices u and v.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Consider three cases to obtain upper bounds on κ(u, v) depending

• n the possible locations of distinct vertices u and v.

◮ Case 1: u ∈ S. If P and P′ are distinct internally

disjoint u, v-paths, then both of them must visit V (M) − T immediately after u. κ(u, v)≤2m−t.

M S=V(G)−M T T T’ R R

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Consider three cases to obtain upper bounds on κ(u, v) depending

• n the possible locations of distinct vertices u and v.

◮ Case 1: u ∈ S. If P and P′ are distinct internally

disjoint u, v-paths, then both of them must visit V (M) − T immediately after u. κ(u, v)≤2m−t.

◮ Case 2: u, v ∈ T ′. κ(u, v) ≤ n − 1 = 2m + s − 1.

M S=V(G)−M T T T’ R R

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. Consider three cases to obtain upper bounds on κ(u, v) depending

• n the possible locations of distinct vertices u and v.

◮ Case 1: u ∈ S. If P and P′ are distinct internally

disjoint u, v-paths, then both of them must visit V (M) − T immediately after u. κ(u, v)≤2m−t.

◮ Case 2: u, v ∈ T ′. κ(u, v) ≤ n − 1 = 2m + s − 1. ◮ Case 3: u ∈ R ∪ T. For the vertex after u on a

u, v-path, at most one vertex of S is available. Thus, κ(u, v) ≤ 2m.

M S=V(G)−M T T T’ R R

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2) Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2)

≤

(2m−t)((s

2)+st)+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−st

”

(n

2) Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2)

≤

(2m−t)((s

2)+st)+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−st

”

(n

2)

= 2m +

(s−1)(t′

2)−t(s 2)−t2s

(n

2)

≤ 2m − t

• s2+3ts+t−1

n(n−1)

• ≤ 2m.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2)

≤

(2m−t)((s

2)+st)+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−st

”

(n

2)

= 2m +

(s−1)(t′

2)−t(s 2)−t2s

(n

2)

≤ 2m − t

• s2+3ts+t−1

n(n−1)

• ≤ 2m.

To have equality in the last inequality, t = 0 or t = 1.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2)

≤

(2m−t)((s

2)+st)+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−st

”

(n

2)

= 2m +

(s−1)(t′

2)−t(s 2)−t2s

(n

2)

≤ 2m − t

• s2+3ts+t−1

n(n−1)

• ≤ 2m.

To have equality in the last inequality, t = 0 or t = 1. t = 1 requires s = 0. We cannot have equality.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Connectivity and Matching Number)

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G). Thisissharponlyforcomplete graphswithan odd number of vertices. κ(G) ≤

(2m−t)((s

2)+s(n−s))+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−s(n−s)

”

(n

2)

≤

(2m−t)((s

2)+st)+(2m+s−1)(t′ 2)+2m

“

(n

2)−(s 2)−(t′ 2)−st

”

(n

2)

= 2m +

(s−1)(t′

2)−t(s 2)−t2s

(n

2)

≤ 2m − t

• s2+3ts+t−1

n(n−1)

• ≤ 2m.

To have equality in the last inequality, t = 0 or t = 1. t = 1 requires s = 0. We cannot have equality. t = 0 requires s = 1. Thus, G = Kn.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (κ(G) and α′(G) in Bipartite)

Theorem (Kim and O 2011++) If G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G).

This is sharp only for Kq,3q−2 for a positive integer q.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (κ(G) and α′(G) in Bipartite)

Theorem (Kim and O 2011++) If G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G).

This is sharp only for Kq,3q−2 for a positive integer q. Sketch of proof:

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (κ(G) and α′(G) in Bipartite)

Theorem (Kim and O 2011++) If G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G).

This is sharp only for Kq,3q−2 for a positive integer q. Sketch of proof:

◮ Let M be a maximum matching in G with partite

sets A and B. Let A1 = A − V (M) and let B1 = B − V (M).

M A B B A1

1

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (κ(G) and α′(G) in Bipartite)

Theorem (Kim and O 2011++) If G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G).

This is sharp only for Kq,3q−2 for a positive integer q. Sketch of proof:

◮ Let M be a maximum matching in G with partite

sets A and B. Let A1 = A − V (M) and let B1 = B − V (M).

◮ Let A2 be all vertices in A that are reachable by an

M-augmenting path from a vertex in B1, and let B2 be all vertices in B that are reachable by an M-augmenting path from a vertex in A1.

M A B B A1

1

A2 B

2

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (κ(G) and α′(G) in Bipartite)

Theorem (Kim and O 2011++) If G is connected and bipartite, then κ(G) ≤

• 9

8 − 3n−4 8n2−8n

• α′(G).

This is sharp only for Kq,3q−2 for a positive integer q. Sketch of proof:

◮ Let M be a maximum matching in G with partite

sets A and B. Let A1 = A − V (M) and let B1 = B − V (M).

◮ Let A2 be all vertices in A that are reachable by an

M-augmenting path from a vertex in B1, and let B2 be all vertices in B that are reachable by an M-augmenting path from a vertex in A1.

◮ Let A3 = A − (A1 ∪ A2) and B3 = B − (B1 ∪ B2).

M A B B A1

1

A2 B

2

B

3

A3

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Connectivity

Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ(G) ≤ d.

Theorem (Kim and O 2011++) For a connected graph G, κ(G) ≤ 2α′(G), and this is sharp. Furthermore, if G is connected and bipartite, then κ(G) ≤ 9

8α′(G) − 3n−4 8n2−8nα′(G), and this is sharp.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-Connectivity

Theorem (Dankelmann and Oellermann 2003) If G has average degree d and n vertices, then

d

2

n−1 ≤ κ′(G) ≤ d.

Theorem (Kim and O 2011++) For a connected graph G, κ′(G) ≤ 2α′(G), and this is sharp. Furthermore, if G is connected and bipartite, then κ′(G) ≤ 9

8α′(G) − 3n−4 8n2−8nα′(G), and this is sharp.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity

Kq Kq

times

S Figure: κ(G) = 1 + O( q

s ) and κ′(G) = q − 1

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity

Kq Kq

times

S Figure: κ(G) = 1 + O( q

s ) and κ′(G) = q − 1

The above graphs show that there can be a huge gap between average edge-connectivity and average connectivity.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity

Kq Kq

times

S Figure: κ(G) = 1 + O( q

s ) and κ′(G) = q − 1

The above graphs show that there can be a huge gap between average edge-connectivity and average connectivity. An extremal problem: What is the smallest average edge-connecitivity of connected r-regular graphs with n vertices?

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3. Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3. Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in the following family.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3. Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in the following family. If a graph G has a cut-edge, then we get components after we delete all cut-edges of G.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3. Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in the following family. If a graph G has a cut-edge, then we get components after we delete all cut-edges of G. We define an i-balloon to be such a component incident to i-cut-edges.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Average Edge-connectivity and Cubic Graphs

We found the best lower bound for the first nontrivial case r = 3. Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in the following family. If a graph G has a cut-edge, then we get components after we delete all cut-edges of G. We define an i-balloon to be such a component incident to i-cut-edges. Let B1 = P3 + K2 and let B′

1 = K4 − e.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1:

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1:

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1: If not, then there exists an 1-balloon D1 of G such that D1 = B1. Let |V (D1)| = 5 + a.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1: If not, then there exists an 1-balloon D1 of G such that D1 = B1. Let |V (D1)| = 5 + a. Let G ′ be the graph obtained from G by replacing D1 with B1.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1: If not, then there exists an 1-balloon D1 of G such that D1 = B1. Let |V (D1)| = 5 + a. Let G ′ be the graph obtained from G by replacing D1 with B1. Then κ′(G ′) n−a

2

• ≥

n−1

2

• + 7(n−1)+58

4

.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1: If not, then there exists an 1-balloon D1 of G such that D1 = B1. Let |V (D1)| = 5 + a. Let G ′ be the graph obtained from G by replacing D1 with B1. Then κ′(G ′) n−a

2

• ≥

n−1

2

• + 7(n−1)+58

4

.

κ′(G) `n

2

´ = κ′(G ′) `n−a

2

´ −κ′(B1) `5

2

´ −5(n−a−5)+κ′(D1) `5+a

2

´ +(5+a)(n−a−5)

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1: If not, then κ′(G) n

2

• ≥ 2

n

2

• ≥

n

2

• + 7n+58

4

. Every 1-balloon of G is B1: If not, then there exists an 1-balloon D1 of G such that D1 = B1. Let |V (D1)| = 5 + a. Let G ′ be the graph obtained from G by replacing D1 with B1. Then κ′(G ′) n−a

2

• ≥

n−1

2

• + 7(n−1)+58

4

.

κ′(G) `n

2

´ = κ′(G ′) `n−a

2

´ −κ′(B1) `5

2

´ −5(n−a−5)+κ′(D1) `5+a

2

´ +(5+a)(n−a−5) ≥ `n−a

2

´ + 7(n−a)+58

4

−26−5(n−a−5)+2 `5+a

2

´ +(5+a)(n−a−5) > `n

2

´ + 7n+58

4

n

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Proof (Average Edge-connectivity and Cubic Graphs)

Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4, then n

2

• κ′(G) ≥

n

2

• + 7n+58

4

. Equality holds only for graphs in a special family. Sketch of proof: Consider a minimal counterexample G. κ′(G) = 1. Every 1-balloon of G is B1. Every 2-balloon of G is B′

1.

There are no i-balloons in G for i ≥ 3.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have?

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have? 4.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have? 4. Every cubic graph without cut-edges has a perfect matching.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have? 4. Every cubic graph without cut-edges has a perfect matching. Thus, it is natural to ask how many perfect matchings a 2-edge-connected cubic graph must have.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have? 4. Every cubic graph without cut-edges has a perfect matching. Thus, it is natural to ask how many perfect matchings a 2-edge-connected cubic graph must have. In the 1970s, Lov´ asz and Plummer conjectured that if G is a cubic graph without cut-edges, then pm(G) should grow exponentially with the number of vertices of G, where pm(G) is the number of perfect matchings in G.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

How many perfect matchings do the above graphs have? 4. Every cubic graph without cut-edges has a perfect matching. Thus, it is natural to ask how many perfect matchings a 2-edge-connected cubic graph must have. In the 1970s, Lov´ asz and Plummer conjectured that if G is a cubic graph without cut-edges, then pm(G) should grow exponentially with the number of vertices of G, where pm(G) is the number of perfect matchings in G. This was proved by Esperet, Kardos, King, Kr´ al, and Norin.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

Now, if we weaken the condition “2-edge-connectedness” to “has a perfect matching”, then how many perfect matchings must a cubic graph have?

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

Now, if we weaken the condition “2-edge-connectedness” to “has a perfect matching”, then how many perfect matchings must a cubic graph have? The answer is Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4 and has a perfect matching, then pm(G) ≥ 4.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Perfect Matchings in Regular Graphs

Now, if we weaken the condition “2-edge-connectedness” to “has a perfect matching”, then how many perfect matchings must a cubic graph have? The answer is Theorem (Kim and O 2011++) If G is a cubic graph with n vertices, other than K4 and has a perfect matching, then pm(G) ≥ 4. Equality holds for infinitely many graphs.

Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs

Thank you

Thank You : )

Suil O Average Connectivity and Average Edge-connectivity in Graphs