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Nonunicyclic Graphs with Prime Vertex Labelings, II

Dana Ernst
March 20, 2015

Nonunicyclic Graphs with Prime Vertex Labelings, II

This talk was given by my undergraduate research student Alyssa Whittemore on March 20, 2015 at the 2015 MAA/CURM Spring Conference at Brigham Young University.

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

Dana Ernst

March 20, 2015
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  1. Nonunicyclic Graphs with Prime Vertex
    Labelings, II
    Alyssa Whittemore
    Joint work with: Nathan Diefenderfer, Michael Hastings, Levi
    Heath, Hannah Prawzinsky, Briahna Preston & Emily White
    CURM Conference
    March 20, 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.

    View Slide

  3. Infinite Families of Graphs
    P8
    C12
    S5

    View Slide

  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.

    View Slide

  5. 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

  6. 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

  7. Book Generalizations
    Theorem
    All Sn × Pm
    are prime for 3 ≤ m ≤ 7.

    View Slide

  8. Example of Sn × P3
    1 2 3
    6 5 4
    8 9 7
    12 11 10
    14 15 13
    18 17 16
    20 21 19 A

    A

    A


    View Slide

  9. Example of Sn × P4
    1 2 3 4
    6 7 8 5
    12 11 10 9
    16 15 14 13
    18 19 20 17
    24 23 22 21
    28 27 26 25 ←

    A


    A

    View Slide

  10. Example of Sn × P5
    1 2 3 4 5
    10 9 8 7 6
    12 13 14 15 11
    18 19 20 17 16
    24 23 22 25 21
    30 29 28 27 26
    32 33 34 35 31
    38 39 40 37 36 B
    A

    C
    B
    A


    View Slide

  11. Example of Sn × P6
    1 2 3 4 5 6
    12 11 10 9 8 7
    18 17 16 15 14 13
    20 21 22 23 24 19
    30 29 28 27 26 25
    36 35 34 33 32 31
    42 41 40 39 38 37
    48 47 46 45 44 43
    50 51 52 53 54 49 A




    A



    View Slide

  12. Summary of Results
    • Sn × P3
    has a 2-row repeating pattern: ←, A.
    • Sn × P4
    has a 3-row repeating pattern: A, ←, ←.
    • Sn × P5
    has a 6-row repeating pattern: A, B, C, ←, A, B.
    • Sn × P6
    has a 5-row repeating pattern: ←, ←, ←, ←, A.
    The simplicity ends here....

    View Slide

  13. Sn × P7
    We have found a prime vertex labeling for Sn × P7
    .
    There are 10 ordered row permutations. A pattern of ordered
    row permutations begins at row 3 and repeats in blocks of 30 for
    as many rows as needed.

    View Slide

  14. Conjecture
    Conjecture
    Sn × Pm
    is prime for all m ≥ 2 and n ≥ 1.
    Here is an example of Sn × P12
    , specifically 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
    We have found a prime labeling that works for more than
    205,626 rows (2,467,524 vertices) of this family.

    View Slide

  15. Future Work
    Larger Generalized Books?

    View Slide

  16. Acknowledgments
    Center for Undergraduate Research in Mathematics
    Northern Arizona University
    Research Advisors Dana Ernst and Jeff Rushall

    View Slide

  17. Book Generalizations
    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

  18. 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