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).

Transcript

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
   

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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.
   

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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.
   

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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
   .
   

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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
   

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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.
   

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

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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
    

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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
    .
    

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13. Example of C4
    P2
    S4
    

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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.
    

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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
    

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

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

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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
    

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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
    

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20. EXCELing with Computer Aid
    

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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
    

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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.
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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