Investigating Investigating properties ofproperties of

Kneser GraphsKneser Graphs

Modesty BriggsModesty BriggsCalifornia State University, NorthridgeCalifornia State University, Northridge

Sponsored by JPL/NASA Pair program;Sponsored by JPL/NASA Pair program;Funded by NSA and NSFFunded by NSA and NSF

What is a Kneser Graph?What is a Kneser Graph?

For For n n ≥ 2t + 1≥ 2t + 1, , the Kneser graph, the Kneser graph, K( n, t), K( n, t), is the graph whose vertices is the graph whose vertices are the are the tt subsets of an subsets of an nn-set. -set.

Example: K (5, 2)Example: K (5, 2)

[n]=[5] = { 1, 2, 3, 4, 5}[n]=[5] = { 1, 2, 3, 4, 5}

t=2t=2

{1,2}{1,2}

{1,3}{1,3}

{1,4}{1,4}

{1,5}{1,5}{2,4}{2,4}{2,3}{2,3}

{4,5}{4,5}

{3,4}{3,4}{2,5}{2,5}

{3,5}{3,5}

{ 1, 2, 3, 4, 5}{ 1, 2, 3, 4, 5}

What is a Kneser?What is a Kneser?

ForFor n n ≥ 2t + 1≥ 2t + 1, , the Kneser graph, the Kneser graph, K( n, t), K( n, t), is the graph whose verticesis the graph whose vertices are theare the tt subsets of ansubsets of an nn-set.-set.

Vertices are adjacent when Vertices are adjacent when corresponding subsets are disjoint.corresponding subsets are disjoint.

1,2

2,5

2,4 2,3

4,5

1,4

1,3

3,5

1,5

3,4

PETERSON PETERSON GRAPHGRAPH

isis

K (5,2)K (5,2)

1,2

2,5

2,4

2,3

4,5

1,4

1,3

3,5

1,5

3,4

K (7,3)35 vertices

DefinitionsDefinitions

Distance – the length of the shortest path Distance – the length of the shortest path from vertex u to vertex v of a graph.from vertex u to vertex v of a graph.

u v

DefinitionDefinition

Diameter – The longest distance in a graph Diameter – The longest distance in a graph G.G.

u v

Diameter Diameter (2t +1 (2t +1 ≤ n < 3t -1) ≤ n < 3t -1)

Fact Fact

For n For n ≥ 3t -1, ≥ 3t -1, diamdiam(K ( n, t)= 2.(K ( n, t)= 2.

AssumptionAssumption

2 2 << diam diam(K ( n, t) (K ( n, t) ≤≤ t t

Girth Girth ((2t +1 2t +1 ≤ n < 3t -1)≤ n < 3t -1) The length of the shortest cycleThe length of the shortest cycle

Theorem: Let K(n,t) be a Kneser Graph Theorem: Let K(n,t) be a Kneser Graph with n<3t-1. with n<3t-1.

4 4 n > 2t + 1 n > 2t + 1 girth K(n,t) = girth K(n,t) =

6 6n = 2t + 1 n = 2t + 1

WHY NOT 3?WHY NOT 3?

Let A be subset of {1,2,…,n} containing t elements. There exist a B subset of {1,2,…,n}, such that A ∩ B =Ø.

Now consider subset C of {1,2,…,n} containing t elements, such that

A ∩ C=Ø .

Then either B ∩ C=Ø or B ∩ C≠Ø .

Assume BAssume B ∩∩ C = Ø. Then subsets A, B, C are C = Ø. Then subsets A, B, C are mutually disjoint. mutually disjoint.

A

B C

|AυBυC|= |A| + |B| + |C| = t + t + t = 3t ≤ n

Then, 3t ≤ n < 3t -1But 3t < 3t – 1 is a contradiction.Therefore, B ∩ C ≠ Ø and there will

not exist a cycle of length three.

Hence, the girth K( n, t) > 3 when n<3t -1.Hence, the girth K( n, t) > 3 when n<3t -1.

A

B C

Assume n > 2t+1. WLOG, let A be the subset {1,…,t} of n-set.

Since subsets B and C are both disjoint to A, then B and C may be chosen such that |B U C|= t +1(maintaining B ∩ A= C ∩ A=Ø).

So,n> 2t +1= t + (t +1)

= |A| + | B U C | = | A U B U C |

Then n > | A U B U C |.

Therefore, there are elements in n that are not in A, B, or C.

Hence, another subset D can be composed of t elements not in B or C.

{1,…,t}

{t+1,…,2t}

{t+1,…, 2t-1, 2t+1}

{1,…, t-1, 2t+2}

Therefore, cycle length is four. Therefore, cycle length is four. Hence the girth K(n,t)=4 when Hence the girth K(n,t)=4 when

2t+1< n< 3t-12t+1< n< 3t-1

Assume n= 2t+1

As |A|=t and |B U C|= t+1, we have

n= 2t + 1= t + (t+1)

= |A| + |B U C|

= |A U B U C| .

So, n = |A U B U C|

{1,…,t}

{t+1,…,2t}{t+1,…, 2t-1, 2t+1}

{2,…, t, 2t}{2,…,t, 2t+1}

Assume DAssume D ∩∩ E = Ø. E = Ø.

So,|D U E|= |D| + |E|

= t + t = 2t

However, |D U E|= t + 2 when n = 2 t + 1.

Therefore, since t+2 < 2t, D ∩ E≠Ø and there will not be a cycle of length five.

{1,…,t}

{t+1,…,2t}{t+1,…, 2t-1, 2t+1}

{2,…, t, 2t}{2,…,t, 2t+1}

Hence, the girth K( n, t) > 5 Hence, the girth K( n, t) > 5 when 2t + 1<3t -1.when 2t + 1<3t -1.

{1, t+1,…, 2t-1}

{1,…,t}

{t+1,…,2t}{t+1,…, 2t-1, 2t+1}

{2,…, t, 2t}{2,…,t, 2t+1}

What Next ?What Next ?

Will the diameter equal t Will the diameter equal t as n gets closer to as n gets closer to

2t + 1.2t + 1.

Special Thanks

• JPL/NASA PAIR PROGRAM

• NSF and NSA for funding

• Dr. Carol Shubin

• Dr. Cynthia Wyels (CAL Lutheran)

• Dr. Michael Neubauer

• Various Professors in the Math Department