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Nonunicyclic Graphs with Prime Vertex Labelings, I

Dana Ernst
March 20, 2015

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).

Dana Ernst

March 20, 2015
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  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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