New Results on Prime Vertex Labelings, III

This talk was given by my undergraduate research students Michael Hastings, Hannah Prawzinsky, and Alyssa Whittemore on February 28, 2015 at the at the Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of Texas at El Paso.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Transcript

1. New Results on Prime Vertex Labelings, III
   Michael Hastings, Hannah Prawzinsky & Alyssa Whittemore
   Joint work with: Nathan Diefenderfer, Levi Heath, Briahna
   Preston & Emily White
   SUnMaRC
   February 28, 2015
   

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2. What is a Graph?
   Definition
   A graph G(V, E) is a set V of vertices and a set E of edges
   connecting some (possibly empty) subset of those vertices. A
   simple graph is a graph that contains neither “loops” nor
   multiple edges between vertices. A connected graph is a
   graph in which there exists a “path” between every pair of
   vertices.
   For the remainder of the presentation, all graphs are assumed
   to be simple and connected.
   

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3. Infinite Families of Graphs
   P8
   C12
   S5
   

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4. Prime Vertex Labelings
   Definition
   An n-vertex graph has a prime vertex labeling if its vertices are
   labeled with the integers 1, 2, 3, . . . , n such that no label is
   repeated and all adjacent vertices (i.e., vertices that share an
   edge) have labels that are relatively prime.
   1
   6
   7
   4
   9
   2
   3
   10
   11
   12
   5
   8
   Some useful number theory facts:
   All pairs of consecutive integers
   are relatively prime.
   Consecutive odd integers are
   relatively prime.
   A common divisor of two integers
   is also a divisor of their difference.
   The integer 1 is relatively prime to
   all integers.
   

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5. Cycle Chains
   Definition
   A cycle chain, denoted Cm
   n
   , is a graph that consists of m
   different n-cycles adjoined by a single vertex on each cycle
   (each cycle shares a vertex with its adjacent cycle(s)).
   Here we show labelings for Cm
   4
   , Cm
   6
   , and Cm
   8
   . The labelings for
   these three infinite families of graphs all employ similar
   strategies.
   

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6. Example of C4
   8
   

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7. Cycle Chain Results
   Theorem
   All Cm
   8
   are prime using the labeling function: f(ci,k ) = 7i + k − 6
   Theorem
   All Cm
   6
   are prime using the labeling function: f(ci,k ) = 5i + k − 4
   Theorem
   All Cm
   4
   are prime using the labeling function f(ci,k ) = 3i + k − 1
   

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8. Labeled C5
   8
   1
   2
   3
   4
   5
   6
   7
   8
   15
   11
   10
   9
   1
   12
   13
   14
   19
   18
   17
   16
   15
   22
   21
   20
   29
   25
   24
   23
   19
   26
   27
   28
   33
   32
   31
   30
   29
   36
   35
   34
   

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9. Labeled C5
   6
   1
   2
   3
   4
   5 6
   11
   8
   7
   1
   9 10
   16
   13
   12
   11
   14 15
   19
   18
   17
   16
   21 20
   26
   23
   22
   19
   24 25
   

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10. Labeled C4
    4
    5
    4
    3
    2
    7
    6
    5
    8
    11
    9
    7
    10
    13
    12
    11
    1
    

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11. Labeled C5
    4
    5
    4
    3
    2
    7
    6
    5
    8
    11
    9
    7
    10
    13
    12
    11
    14
    1
    15
    13
    16
    

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12. Hastings Helms
    Definition
    A Hastings Helm, denoted Hn
    , is the graph Cn × P2
    .
    Here we will show that if n is odd, then Hn
    is not prime and will
    show that if either n + 1 or n − 1 is prime, then Hn
    has a prime
    vertex labeling.
    The remaining cases are currently open. We conjecture that Hn
    is prime for all even n.
    

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13. Hastings Helms
    Theorem
    If n is odd, then Hn
    is not prime.
    Proof.
    Parity argument.
    1
    5
    4 3
    2
    7
    6
    10 9
    8
    

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14. Hastings Helms
    Theorem
    If n + 1 is prime, then Hn
    is prime.
    Labeling function
    Let c1
    1
    , c1
    2
    , . . . , c1
    n
    denote the vertices on the inner cycle, and
    c2
    1
    , c2
    2
    , ..., c2
    n
    be the vertices on the outer cycle. The labeling
    formula f : V → {1, 2, . . . , 2n} is given by:
    f(cj
    i
    ) =
    
    
    
    
    
    i, i = 1, 2, . . . , n, j = 1
    i + n + 1, i = 1, 2, . . . , n − 1, j = 2
    n + 1, i = n, j = 2
    

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15. Example of H6
    (Hastings Helms)
    1
    2 3
    4
    5
    6
    8
    9 10
    11
    12
    7
    

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16. Hastings Helms
    Theorem
    If n − 1 is prime, then Hn
    is prime.
    Labeling function
    Let c1
    1
    , c2
    1
    , . . . , cn
    1
    denote the vertices on the ”inner” cycle and
    c1
    2
    , c2
    2
    , . . . , cn
    2
    denote the corresponding vertices on the ”outer”
    cycle. The labeling formula f : V → {1, 2, . . . , 2n} is given by:
    f(ci
    1
    ) =
    
    
    
    
    
    
    
    
    
    i, i = 2, 3, . . . n − 2
    n − 1, i = 1
    1, i = n − 1
    2n, i = n
    f(ci
    2
    ) =
    i + n − 1, i = 2, 3, . . . n
    n, i = 1
    

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17. Example of H8
    (Hastings Helms)
    5
    4
    3
    2
    1
    16
    7
    6
    12
    11
    10
    9
    8
    15
    14
    13
    

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18. Example of H8
    (Hastings Helms)
    5
    4
    3
    2
    7
    16
    1
    6
    12
    11
    10
    9
    8
    15
    14
    13
    

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19. General labeling for Hn
    when n − 1 is prime
    n − 1 2
    n − 2
    1
    2n
    n n + 1
    2n − 3
    2n − 2
    2n − 1
    

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20. Books
    Definition
    A book is the graph Sn × P2
    , where Sn
    is the star with n pendant
    vertices and P2
    is the path with 2 vertices.
    Here is a picture of S4 × P2
    :
    3
    5
    7
    9
    4
    6
    8
    10
    2 1
    It is known that all books have a prime labeling.
    

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21. Book Generalizations
    Definition
    A generalized book is a graph of the form Sn × Pm
    , which looks
    like m − 1 books glued together.
    Here is a picture of S5 × P4
    :
    

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22. Book Generalizations
    Theorem
    All Sn × P4
    are prime.
    Here is an example of S5 × P4
    :
    24 5 6 7
    8 9 10 11
    12 13 14 15
    16 17 18 19
    20 21 22 3
    23 2 1 4
    

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23. Book Generalizations
    Theorem
    All Sn × P6
    are prime.
    

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24. Book Generalizations
    Labeling
    Let vi,n
    be the vertex in the ith star and the nth row, and let
    c1, c2, c3, c4, c5, c6
    form a path through the center of each Sn
    .
    Then our labeling function f : V → {1, 2, . . . , 6(n + 1)} is given by:
    f(c1) = 6, f(c2) = 1, f(c3) = 2, f(c4) = 3, f(c5) = 4,
    and f(c6) = p, the largest prime ≡ 1, 7, 13, 17, or 23 (mod 30).
    Note that p = 6t + s where s = 1, 5. Let f(vs,t ) = 5. Finally,
    f(v1,1) = 7
    f(v2,1) = 8
    f(v3,1) = 9
    f(v4,1) = 10
    f(v5,1) = 11
    f(v6,1) = 12
    f(vi,k ) = vi,k−1 + 6 for 1 ≤ i ≤ 6 and 2 ≤ k ≤ n
    

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25. Book Generalizations
    Here is an example of the prime labeling for Sn × P6
    , in
    particular, S4 × P6
    :
    7 8 9 10 11 12
    13 14 15 16 17 18
    19 20 21 22 5 24
    25 26 27 28 29 30
    6 1 2 3 4 23
    

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26. Conjecture
    Conjecture
    Sn × P12
    is prime.
    We have found a prime labeling that works for more than
    205,626 rows (2,467,524 vertices).
    Here is an example of S3 × P12
    :
    13 14 15 16 17 18 19 20 21 22 23 24
    25 26 27 28 29 30 31 10 33 34 35 36
    37 38 39 40 11 42 5 44 45 46 7 48
    2 3 4 1 6 43 8 9 32 47 12 41
    

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27. Future Work
    Larger Generalized Books?
    Other Hastings Helms?
    Cycle chains with larger or odd cycles?
    

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28. Acknowledgments
    SUnMaRC Organizers
    Center for Undergraduate Research in Mathematics
    Northern Arizona University
    Office of Undergraduate Research, NAU
    Research Advisors Dana Ernst and Jeff Rushall
    

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