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New Results on Prime Vertex Labelings, I

Dana Ernst
February 28, 2015

New Results on Prime Vertex Labelings, I

This talk was given by my undergraduate research students Levi Heath and Emily White 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, I
    Levi Heath and Emily White
    Joint work with: Nathan Diefenderfer, Michael Hastings,
    Hannah Prawzinsky, Briahna Preston & Alyssa Whittemore
    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.

    View Slide

  3. Simple Graphs
    Definition
    A simple graph is a graph that contains neither “loops” nor
    multiple edges between vertices.
    For the rest of the presentation, all graphs are assumed to be
    simple. Here is a graph that is NOT simple.

    View Slide

  4. Connected and Unicyclic Graphs
    Definition
    A connected graph is a graph in which there exists a “path”
    between every pair of vertices.
    For the rest of the presentation, all graphs are assumed to be
    connected.
    Definition
    A unicyclic graph is a simple graph containing exactly one
    cycle.
    Here is a unicyclic graph that is NOT connected.

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  5. Pendants
    Definition
    In a unicyclic graph, a pendant is a path on two vertices with
    exactly one vertex being a cycle vertex. The non-cycle vertex
    of a pendant is called a pendant vertex.

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  6. Infinite Families of Graphs
    P8
    C12
    S5

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  7. Gluing Function
    Definition
    G1
    G2
    is the graph that results from ”selectively gluing” copies
    of G2
    to some vertices of G1
    .
    Here is C3
    P2
    S3
    .

    View Slide

  8. Graph Labelings
    Definition
    A graph labeling is an “assignment” of integers (possibly
    satisfying some conditions) to the vertices, edges, or both.
    Formal graph labelings, as you will soon see, are functions.
    2 3 2 3
    1 4 1 4
    1
    2
    3
    4
    1
    2 3
    4

    View Slide

  9. Prime Vertex Labeling
    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

  10. Conjecture
    Conjecture (Seoud and Youssef, 1999)
    All unicyclic graphs have a prime vertex labeling.

    View Slide

  11. Known Prime Vertex Labelings
    1 2 3 4 5 6 7 8
    P8
    1
    12
    11
    10
    9
    8
    7
    6
    5
    4
    3
    2
    C12
    1
    2
    6
    5
    4 3
    S5

    View Slide

  12. New Infinite Families of Graphs
    Definition
    A cycle pendant star is a cycle with each cycle vertex
    adjacent to an identical star, denoted Cn
    P2
    Sk
    .
    Definition
    A generalized cycle pendant star is a cycle with each cycle
    vertex adjacent to an identical star and each noncentral star
    vertex is a central vertex for an identical star, denoted
    Cn
    P2
    Sk
    Sk
    Sk · · · Sk
    .

    View Slide

  13. Example of C4
    P2
    S4

    View Slide

  14. Prime Vertex Labelings of Infinite Families of
    Graphs
    Theorem
    For all n ∈ N, we have:
    The cycle pendant stars, Cn
    P2
    Sk
    are prime for 3 ≤ k ≤ 8.
    The generalized cycle pendant star, Cn
    P2
    S3
    S3
    has a
    prime vertex labeling.
    The cases for k ≥ 9 for cycle pendant stars and all other cases
    of generalized cycle pendant stars remain open.

    View Slide

  15. Labeling Function of Cn
    P2
    S3
    Labeling Function
    Let c1, c2, . . . , cn
    denote the cycle labels, p1, p2, . . . , pn
    denote
    the vertices adjacent to the corresponding cycle vertices and
    the pendant vertices adjacent to pi
    be denoted si,j, 1 ≤ j ≤ 3.
    The labeling function f : V → {1, 2, . . . , 5n} is given by:
    f(ci) = 5i − 4, 1 ≤ i ≤ n
    f(pi) =
    5i − 2, if i is odd
    5i − 3, if i is even
    f(si,j) =





    5i − 3 + j, i is even
    5i − 2 + j, j = 3 and i is odd
    5i − 3, j = 3 and i is odd

    View Slide

  16. Prime Vertex Labeling of Cn
    P2
    S3
    1
    3
    4
    2 5
    11
    13
    14
    12
    15
    6
    7
    9
    8
    10

    View Slide

  17. Generalized Labeling of Cn
    P2
    S4
    6i − 5 6i − 1
    6i − 4
    6i − 3
    6i − 2
    6i
    6i − 11
    6i + 1

    View Slide

  18. Prime Vertex Labeling of C4
    P2
    S4
    1
    5
    2
    3
    4
    6
    19
    23
    20
    21
    22
    24
    7
    11
    8
    9
    10
    12
    13
    17
    14
    15
    16
    18

    View Slide

  19. Prime Vertex Labeling of C5
    P2
    S6
    1
    5
    2
    3 4 6 7
    8
    9
    13
    10
    11
    12
    14
    15
    16
    17
    19
    18
    20
    21
    22
    23
    24
    25
    29
    26
    27
    28
    30
    31
    32
    33
    37
    34
    35
    36
    38
    39
    40

    View Slide

  20. EXCELing with Computer Aid

    View Slide

  21. Prime Vertex Labeling of C4
    P2
    S7
    1
    5
    2
    3
    4
    6
    7
    8
    9
    10
    13
    11
    12
    14
    15
    16
    17
    18
    19
    23
    20
    21
    22
    24
    25
    26
    27
    28
    31
    29
    30
    32
    33
    34
    35
    36

    View Slide

  22. Prime Vertex Labeling of Cn
    P2
    S3
    S3
    1
    2
    5
    9
    11
    3 4
    6
    7
    8
    10
    12
    13
    14
    15
    16
    19
    23
    25
    17
    18
    20
    21
    22
    24
    26
    27
    28
    29
    32
    31
    35
    41
    30
    33
    34
    36
    37
    38
    39
    40
    42
    43
    44
    47
    51
    53
    45
    46
    48
    49
    50
    52
    54
    55 56

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  23. Future Work
    Conjecture
    For all n, k ∈ N, Cn
    P2
    Sk
    is prime.
    Conjecture
    For all n, k ∈ N, Cn
    P2
    Sk
    Sk
    Sk · · · Sk
    is prime.

    View Slide

  24. 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

  25. Labeling Function of Cn
    P2
    S3
    S3
    Note that Cn
    P2
    S3
    S3
    contains 14n vertices. We will identify
    our vertices as follows. Let ci, 1 ≤ i ≤ n denote the cycle
    vertices, let pi
    denote the pendant vertex adjacent to ci
    , let
    the non-cycle vertices adjacent to pi
    be denoted si,j
    for
    1 ≤ j ≤ 3, and let the remaining vertices adjacent to each si,j
    be denoted li,j,k
    for 1 ≤ k ≤ 3. Our labeling function
    f : V → {1, 2, . . . , 14n} is best defined by first describing cycle
    and pendent vertex labels:
    f(ci) = 14i − 13, 1 ≤ i ≤ n
    f(pi) =
    14i − 12, i ≡3
    1, 2
    14i − 10, i ≡3
    0

    View Slide

  26. Labeling Function of Cn
    P2
    S3
    S3
    The remaining vertex labels are determined by the values of i, j
    and k as follows. If i ≡3
    1, 2, then define
    f(si,j) =





    14i − 9, j = 1
    14i − 5, j = 2
    14i − 3, j = 3
    f(li,j,k ) =

































    14i − 11, j = 1, k = 1
    14i − 10, j = 1, k = 2
    14i − 8, j = 1, k = 3
    14i − 7, j = 2, k = 1
    14i − 6, j = 2, k = 2
    14i − 4, j = 2, k = 3
    14i − 2, j = 3, k = 1
    14i − 1, j = 3, k = 2
    14i, j = 3, k = 3

    View Slide

  27. Labeling Function of Cn
    P2
    S3
    S3
    If i ≡3
    0, then define
    f(si,j) =





    14i − 11, j = 1
    14i − 7, j = 2
    14i − 1, j = 3
    f(li,j,k ) =

































    14i − 12, j = 1, k = 1
    14i − 9, j = 1, k = 2
    14i − 8, j = 1, k = 3
    14i − 6, j = 2, k = 1
    14i − 5, j = 2, k = 2
    14i − 4, j = 2, k = 3
    14i − 3, j = 3, k = 1
    14i − 2, j = 3, k = 2
    14i, j = 3, k = 3

    View Slide

  28. Labeling Function of Cn
    P2
    S4
    Let ci
    , 1 ≤ i ≤ n, denote the cycle vertices, let pi
    denote the
    pendant vertex adjacent to ci
    , and let oik
    , 1 ≤ k ≤ 4, denote
    the outer vertices adjacent to pi
    . The labeling function
    f : V → {1, 2, . . . 6n} is given by:
    f(ci) = 6i − 5, 1 ≤ i ≤ n
    f(pi) = 6i − 1, 1 ≤ i ≤ n
    f(oi1) = 6i − 2, 1 ≤ i ≤ n
    f(oi2) = 6i − 3, 1 ≤ i ≤ n
    f(oi3) = 6i − 4, 1 ≤ i ≤ n
    f(oi4) = 6i, 1 ≤ i ≤ n

    View Slide

  29. Labeling Function of Cn
    P2
    S5
    Let ci
    , 1 ≤ i ≤ n, denote the cycle vertices, let pi
    denote the
    pendant vertex adjacent to ci
    , and let oik
    , 1 ≤ k ≤ 5, denote
    the outer vertices adjacent to pi
    . The labeling function
    f : V → {1, 2, . . . 7n} is given by:
    f(ci) = 7i − 6, 1 ≤ i ≤ n
    f(pi) =















    7i − 2, i ≡6
    1, 3
    7i − 3, i ≡6
    2, 4
    7i − 4, i ≡6
    5
    7i − 5, i ≡6
    0, i ≡30
    0
    7i − 1, i ≡30
    0

    View Slide

  30. Labeling Function of Cn
    P2
    S5
    f(oi1) =
    7i − 5, i ≡6
    0 or i ≡30
    0
    7i − 4, i ≡6
    0, i ≡30
    0
    f(oi2) =
    7i − 4, i ≡6
    0, 5 or i ≡30
    0
    7i − 3, i ≡6
    5 or i ≡6
    0, i ≡30
    0
    f(oi3) =
    7i − 3, i ≡6
    0, 2, 4, 5 or i ≡30
    0
    7i − 2, i ≡6
    1, 3
    f(oi4) =
    7i − 2, i ≡30
    0
    7i − 1, i ≡30
    0
    f(oi5) = 7i, 1 ≤ i ≤ n

    View Slide

  31. Labeling Function of Cn
    P2
    S6
    Let ci
    , 1 ≤ i ≤ n, denote the cycle vertices, let pi
    denote the
    pendant vertex adjacent to ci
    , and let oik
    , 1 ≤ k ≤ 6, denote
    the outer vertices adjacent to pi
    . The labeling function
    f : V → {1, 2, . . . 8n} is given by:
    f(ci) = 8i − 7, 1 ≤ i ≤ n
    f(pi) =





    8i − 3, i ≡3
    0
    8i − 5, i ≡3
    0, i ≡15
    0
    8i − 1, i ≡15
    0

    View Slide

  32. Labeling Function of Cn
    P2
    S6
    f(oi1) = 8i − 6, 1 ≤ i ≤ n
    f(oi2) =
    8i − 5, i ≡3
    0 or i ≡15
    0
    8i − 3, i ≡3
    0, i ≡15
    0
    f(oi3) = 8i − 4, 1 ≤ i ≤ n
    f(oi4) =
    8i − 3, i ≡15
    0
    8i − 1, i ≡15
    0
    f(oi5) = 8i − 2, 1 ≤ i ≤ n
    f(oi6) = 8i, 1 ≤ i ≤ n

    View Slide

  33. Labeling Function of Cn
    P2
    S7
    Let ci
    , 1 ≤ i ≤ n, denote the cycle vertices, let pi
    denote the
    pendant vertex adjacent to ci
    , and let oik
    , 1 ≤ k ≤ 7, denote
    the outer vertices adjacent to pi
    . The labeling function
    f : V → {1, 2, . . . 9n} is given by:
    f(ci) = 9i − 8, 1 ≤ i ≤ n
    f(pi) =









    9i − 4, i ≡2
    1
    9i − 5, i ≡2
    0, i ≡10
    0
    9i − 7i ≡10
    0, i ≡70
    0
    9i − 1, i ≡70
    0

    View Slide

  34. Labeling Function of Cn
    P2
    S7
    f(oi1) =
    9i − 7, i ≡10
    0 or i ≡70
    0
    9i − 5, i ≡10
    0, i ≡70
    0
    f(oi2) = 9i − 6, 1 ≤ i ≤ n
    f(oi3) =
    9i − 5, i ≡2
    1 or i ≡70
    0
    9i − 4, i ≡2
    0, i ≡70
    0
    f(oi4) =
    9i − 4, i ≡70
    0
    9i − 1, i ≡70
    0
    f(oi5) = 9i − 3, 1 ≤ i ≤ n
    f(oi6) = 9i − 2, 1 ≤ i ≤ n
    f(oi7) = 9i, 1 ≤ i ≤ n

    View Slide

  35. Labeling Function of Cn
    P2
    S8
    Let ci
    , 1 ≤ i ≤ n, denote the cycle vertices, let pi
    denote the
    pendant vertex adjacent to ci
    , and let oik
    , 1 ≤ k ≤ 8, denote
    the outer vertices adjacent to pi
    . The labeling function
    f : V → {1, 2, . . . 10n} is given by:
    f(ci) = 10i − 9, 1 ≤ i ≤ n
    f(pi) =





    10i − 3, i ≡3
    0
    10i − 7, i ≡3
    0, i ≡21
    0
    10i − 1, i ≡21
    0

    View Slide

  36. Labeling Function of Cn
    P2
    S8
    f(oi1) = 10i − 8, 1 ≤ i ≤ n
    f(oi2) =
    10i − 7, i ≡21
    0 or i ≡3
    1, 2
    10i − 3, i ≡3
    0, i ≡21
    0
    f(oi3) = 10i − 6, 1 ≤ i ≤ n
    f(oi4) = 10i − 5, 1 ≤ i ≤ n
    f(oi5) = 10i − 4, 1 ≤ i ≤ n
    f(oi6) =
    10i − 3, i ≡21
    0
    10i − 1, i ≡21
    0
    f(oi7) = 10i − 2, 1 ≤ i ≤ n
    f(oi8) = 10i, 1 ≤ i ≤ n

    View Slide