Nonunicyclic Graphs with Prime Vertex Labelings, I

This talk was given by my undergraduate research students Michael Hastings & Hannah Prawzinsky 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
   Michael Hastings & Hannah Prawzinsky
   Joint work with: Nathan Diefenderfer, Levi Heath, Briahna
   Preston, Emily White & Alyssa Whittemore
   2015 MAA/CURM Spring 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. 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. Mersenne Primes
    Definition
    A Mersenne prime is a prime number of the form Mn = 2n − 1.
    There are 48 known Mersenne primes. The first few Mersenne
    primes are:
    M2 = 22 − 1 = 3
    M3 = 23 − 1 = 7
    M5 = 25 − 1 = 31
    

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13. Theorem
    All Cm
    n
    , where n = 2k and 2k − 1 is a Mersenne prime, have
    prime labelings.
    

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14. Fibonacci Chains
    Fibonacci sequence
    The sequence, {Fn}, of Fibonacci numbers is defined by the
    recurrence relation Fn = Fn−1 + Fn−2
    , where F1 = 1 and F2 = 1.
    1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
    Proposition
    Any two consecutive Fibonacci numbers in the Fibonacci
    sequence are relatively prime.
    Theorem
    Fibonacci Chains, denoted Cn
    F
    , are prime for all n ∈ N where n is
    the number of cycles that make up the Fibonacci chain.
    

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15. Fibonacci Chains (C5
    F
    )
    1
    2
    4
    3
    5
    6
    7 10
    9
    8
    12
    11
    13
    14
    15
    16
    17
    18
    19
    20
    21
    

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16. Prisms
    Definition
    A Prism, 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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17. Prisms
    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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18. Prisms
    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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19. Example of H6
    (Prisms)
    1
    2 3
    4
    5
    6
    8
    9 10
    11
    12
    7
    

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20. Prisms
    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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21. Example of H8
    (Prisms)
    5
    4
    3
    2
    1
    16
    7
    6
    12
    11
    10
    9
    8
    15
    14
    13
    

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

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23. Example of H32
    (Prisms)
    17
    16
    15
    14
    13
    12
    11
    10
    9
    8
    7
    6
    5
    4
    3
    2
    31
    64
    1
    30
    29
    28
    27
    26 25 24
    23
    22
    21
    20
    19
    18
    48
    47
    46
    45
    44
    43
    42
    41
    40
    39
    38
    37
    36
    35
    34
    33
    32
    63
    62
    61
    60
    59
    58
    57 56 55
    54
    53
    52
    51
    50
    49
    

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24. 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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25. Future Work
    Cycle chains with larger or odd cycles?
    Other cases for Prisms?
    Generalized Prisms?
    

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

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