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

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

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  1. Selected Solutions to Some of Sivaram’s • Suppositions • SubgraphSummability Miriam Larson-Koester, Mount Holyoke College, MA Richard Ligo, Westminster College, PA

  2. What is a graph?

  3. 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

  4. 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

  5. 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

  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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. The S & T Lemma • Lemma: There are three fundamental types of redundancies: • 1) Split redundancies • > Implies 1 redundancy • 2) Adjacent redundancies • > 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

  15. 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

  16. 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

  17. 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)

  18. 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

  19. 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

  20. 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

  21. 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)

  22. 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

  23. 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

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