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).

Transcript

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
   

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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. 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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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
   :
   

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

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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
   ←
   →
   

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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
   →
   

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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
    ←
    →
    

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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
    ←
    ←
    →
    

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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....
    

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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.
    

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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.
    

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15. Future Work
    Larger Generalized Books?
    

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

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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
    

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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
    

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