Miriam Larson-Koester, Mount Holyoke College, MA Richard Ligo, Westminster College, PA

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Selected Solutions to Some of Sivaram’s. Suppositions. Subgraph Summability. Miriam Larson-Koester, Mount Holyoke College, MA Richard Ligo, Westminster College, PA. What is a graph?. What is a graph?. Simple, non-directed graphs Connected induced subgraphs

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Selected Solutions to Some of Sivaram’s

• Suppositions
• SubgraphSummability

Miriam Larson-Koester, Mount Holyoke College, MA

Richard Ligo, Westminster College, PA

What is a graph?
• Simple, non-directed graphs
• Connected induced subgraphs
• A subgraph H of G is said to be induced if for any pair of vertices x and y of H, xy is an edge of H if and only if xy is an edge in G

1

1

1

1

4

1

3

α

G

2

2

1

1

1

1

2

7

1

1

2

5

Introduction to the Problem

3

3

3

3

• This is a 7-labeling, so we say N = 7

1

2

1

2

3

6

1

• However, the subgraphsummability number = 10

3

1

2

4

1

1

3

2

7

2

4

β

G

3

3

1

1

4

4

3

2

10

1

3

8

Introduction to the Problem

4

3

4

4

• This is a 10-labeling, so we say N = 10

3

2

5

3

2

9

1

• The subgraphsummability number = 10

4

3

2

6

Previous Work

• Stephen Penrice and CMU REU programs (2002, 2003, and 2009) have worked extensively on this problem, developing labelings and bounds for numerous graphs and families of graphs

Star

Cycle

Complete

Path

Crown

Complete Bipartite

Broom

The Summability Equation

N + r(α) + e(α) = c(G)

Suppose α is an N-labeling of G

r(α) = the number of redundant label sums in α

e(α) = the number of excess label sums in α

c(G) = the number of connected induced subgraphs of G

The Summability Equation

3

1

N + r(α) + e(α) = c(G)

4

8

1

13

Consider the graph:

6

2

1

1

3

1

1

4

6

2

3

9

11

2

6

2

6

3

9

6

2

3

3

3

1

6

6

3

1

12

1

3

2

3

1

10

6

2

2

6

2

8

6

Sharp Labelings

• If a labeling α has both an excess and redundancy of zero, then α is called sharp

4

1

N + r(α) + e(α) = c(G)

0

0

13

13

• A sharp labeling!

Consider the graph:

6

2

1

1

4

1

1

5

6

2

4

9

12

2

6

2

6

4

10

6

2

1

3

4

1

7

6

4

1

13

2

2

4

1

11

6

2

4

4

6

2

8

6

Work on Cycles

• Do they have sharp labelings?
• Cycles have (n-1)(n) + 1 connected induced subgraphs
• Sharp labelings are known for C3, C4, C5, C6, C8, C9, and C10

1

4

1

1

5

3

4

2

6

2

2

10

• It is known that sharp labelings exist for infinitely many Cn

Difference Sets

• A difference set is essentially a subset D of a group G, the differences of whose elements produce every element of G
• Difference sets allow us to obtain sharp labelings for Cn+1 when n is a prime power

Consider the difference set D = {0, 1, 3, 9} in ℤ13

1

0-9 =

4

= 1-0

9-3 =

= 3-1

2

6

The Mysterious 7-Cycle

• Does it have a sharp labeling?
• Clearly, there does not exist a prime power n such that 7 = n+1
• As a result, difference sets cannot provide the solution we seek…
• C7 has 43 connected induced subgraphs
• Penrice claims that the summability number of C7is 39
• Our computer searches have shown 39 to be the largest labeling for C7
• The most haunting question: What makes the 7-cycle special???

1

8

3

• 39-Labeling

2

14

5

6

The Mysterious 7-Cycle

• We believe that C7 does not have a 40-labeling
• Our computer searches checked every 7-cycle with vertex labels between 1 and 45
• If an (n-1)-labeling does not exist, then an n-labeling does not exist
• A 40-labeling would contain three redundancies if there was no excess
• This led to our creation of the S & T lemma

The S & T Lemma

• Lemma: There are three fundamental types of redundancies:
• 1) Split redundancies
• > Implies 1 redundancy
• > Implies 2 redundancies
• 3) Separate redundancies
• > Implies 4 redundancies

S

N

T

S

T

S

T

M

M

α(S)=α(T)

α(S)+α(M)=α(T)+α(M) α(S)+α(N)=α(T)+α(N)

α(S)+α(M)+α(N)=α(T)+α(M)+α(N)4 redundancies

α(S)=α(T)

α(S)+α(M)=α(T)+α(M)

2 redundancies

α(S)=α(T)

1 redundancy

Lower Bounds for the Summability Number of Cn

• We began with Raj Doshi’s (2003 CMU REU) path labeling system
• Applied to cycles to find new lower bounds
• Found to be the best known lower bounds

2

2

1

1

1

38

2

1

38

10

• Penrice’s Labeling System
• Doshi’s
• Labeling System

83

Labeling

77 Labeling

1

2

6

1

1

7

7

12

1

7

7

12

Centipede Graphs

• They do not appear to have a sharp labeling
• We have counted the number of connected induced subgraphs to be 6∙2n-n-5
• We have developed a system that provides a (5∙2n-6)-labeling

1

1

10

20

5

5

10

20

2

Circulant Graphs

Let S be a subset of {1,…,n}. A circulant graph Cn(S) is a graph with the vertex set {v1, v2,…,vn} and edge set {{vi,vj}=|i-j|∈S}

C6(1,2)

C9(2,4)

Areas of Future Exploration

• Prove that best labeling for C7 is a 39-labeling
• Prove there is no sharp labeling for centipedes
• Explore labelings for multipartite graphs
• Explore more classes of circulant graphs

Tripartite

C6(1,3)

Centipede

Acknowledgments
• Central Michigan University
• Dr. Sivaram Narayan, Central Michigan University
• David Cochran, University of Texas
• Dr. Jordan Webster, MMCC
• Westminster Mathematics Faculty
• Westminster Drinko Center
• NSF REU Grant DMS 08-51321
Bibliography

Penrice, Stephen, Some new graph labeling problems: a preliminary report, DIMACS, preprint, 1995

Narayan, Sivaram, Russell, Janae, Smith, Ken, The subgraph summability number of a biclique, Congressus Numerantium,171, pp 3-11 ,2004

Doshi, Raj, The subgraph summability number of squids and paths, CMU REU, 2003

Maximal Circulant Graphs

• Circulant graphs of the form Cn(1,…, ⌊n/2⌋-1), denoted Xn
• A maximal circulant graph has connected induced subgraphs
• c(Xn)=2n - n/2 -1 if n is even c(Xn)=2n- 2n -1 if n is odd
• Lower bounds for σ(Xn) are also based on parity
• σ(Xn) ≥ (5∙2n-2)/6 if n is even σ(Xn) ≥ 5∙2n-3+1 if n is odd

20

20

1

5

1

5

3

2

3

2

10

40

10

40

The complement of X7

X7 with a 81-labeling

X7 = C7(1,2)

Star Circulant Graphs

• Circulant graphs of the form Cn(2,…, ⌊n/2⌋), denoted Sn
• Lower bounds for σ (Sn) are based on the complement
• σ(Sn) ≥ 5∙2n-3+1 for all n

1

40

2

1

2

40

20

3

20

3

10

5

The complement of S7

S7 with a 81-labeling

S7 = C7(2,3)

10

5

Do All Graphs Have a Sharp Labeling?

Penrice has shown that paths of length ≥ 4 do not have a sharp labeling, and we have proved that a sharp labeling does not exist for complete multipartite graphs

Proposition: Complete bipartite graphs do not have a sharp labeling.

Consider Kr,s with r,s ≥ 3. Call two vertices in R a and b.

Assume that there is an N-labeling for Kr,s and a+b < N.

Then a+b must occur as a sum of some connected induced subgraph.

There are 3 cases:

a+b = c1+c2+…+ck

1)

2)

3)

c1

a

k

a

a+b

a

c2

b

b

k

b

ck

ci

a+b

j

j

k

a+b+k+j=

(a+b)+k +j

a+b+k+j=

(c1+…+ck)+k +j

a+b+k=(a+b)+k