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New Results on Prime Vertex Labelings, III

Dana Ernst
February 28, 2015

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).

Dana Ernst

February 28, 2015
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  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

    View Slide

  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

    View Slide

  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

    View Slide

  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.

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

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

    View Slide

  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

    View Slide

  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.

    View Slide

  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
    :

    View Slide

  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

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

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

    View Slide

  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

    View Slide