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Prime Vertex Labelings of Graphs

Dana Ernst
December 05, 2014

Prime Vertex Labelings of Graphs

This talk was given by my undergraduate research students Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, and Alyssa Whittemore on December 5, 2014 at the Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) at Northern Arizona University.

This research was supported by the National Science Foundation grant #DMS-1148695 through the Center for Undergraduate Research (CURM).

Dana Ernst

December 05, 2014
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  1. Prime Vertex Labelings of Graphs
    Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah
    Prawzinsky, Briahna Preston, Emily White, & Alyssa
    Whittemore
    FAMUS
    December 5, 2014

    View Slide

  2. Why are we here?
    CURM (Center for Undergraduate Research in Math)
    Mini-grant
    MAT 485
    Conferences
    NCUWM - 1/24 (Lincoln, NE)
    SUnMaRC - 2/27 (El Paso, TX)
    CURM Conference - 3/20 (Provo, UT)

    View Slide

  3. 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

  4. 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

  5. Connected 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.
    Here is a graph that is NOT connected.

    View Slide

  6. Cycles
    Definition
    A cycle is a connected path consisting of a single “circuit,” and
    hence has equal numbers of vertices and edges. We let Cn
    denote an n-vertex cycle.
    Below is an image of C12
    .

    View Slide

  7. Unicyclic Graphs
    Definition
    A unicyclic graph (simple and connected) is a graph
    containing exactly one cycle.
    Definition
    G1
    G2
    is the graph that results from “selectively gluing” copies
    of G2
    to some vertices of G1
    .

    View Slide

  8. Example
    Here is C3
    S5
    .

    View Slide

  9. 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

  10. Prime Vertex Labelings
    Definition
    A graph with n vertices has a prime vertex labeling if its vertices
    are labeled with the integers 1, 2, 3, . . . , n and every pair of
    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 Labeling
    Use integers 1–12
    Adjacent vertices must have
    relatively prime labels
    No two vertices can have
    identical labels

    View Slide

  11. Conjecture
    Conjecture (Seoud)
    All unicyclic graphs have a prime vertex labeling.

    View Slide

  12. Cycles and Prime Vertex Labelings
    Theorem
    All cycles have a prime vertex labeling.
    1
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12

    View Slide

  13. More Examples
    Here are examples of infinite families of graphs with prime
    vertex labelings.
    1 2 3 4 5 6 7 8
    16 15 14 13 12 11 10 9
    1 2 3 4 5 6 7 8

    View Slide

  14. More Examples (continued)
    1
    2
    3
    4
    5 6
    1
    11
    9
    7
    5 3
    2
    12
    10
    8
    6 4
    The following presenters will discuss some of their original
    findings for specific families of graphs.

    View Slide

  15. Ternary Trees
    Definition
    A complete ternary tree is a directed rooted tree with every
    internal vertex having 3 children.
    Here are examples of 1 and 2-level complete ternary trees:
    a
    b
    c
    d a
    b
    c
    d
    b1
    b2
    b3
    c1 c2
    c3
    d1
    d2
    d3

    View Slide

  16. New Results
    Theorem 1
    Every Cn
    P2
    S4
    has a prime vertex labeling.
    Theorem 2
    Every Cn
    P2
    S4
    S4
    has a prime vertex labeling.

    View Slide

  17. Theorem 1 Labeling
    Let c1, c2, . . . , cn
    denote the cycle labels, let p1, p2, . . . , pn
    denote the vertices adjacent to the corresponding cycle
    vertices and let the pendant vertices off of 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

  18. Example of Theorem 1 Labeling
    1
    3
    4
    2 5
    11
    13
    14
    12
    15
    6
    7
    9
    8
    10

    View Slide

  19. Cycle Pendant Stars
    Definition
    A star is a graph with one vertex adjacent to all other vertices,
    each of which has degree one.
    Definition
    A cycle pendant star is a cycle with each cycle vertex
    adjacent to an identical star, denoted Cn
    P2
    Sk
    .

    View Slide

  20. Example C4
    P2
    S5

    View Slide

  21. Cn
    P2
    S5
    Theorem
    Cn
    P2
    S5
    is prime for all n.
    Labeling
    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

  22. General Labeling
    6i − 5 6i − 1
    6i − 2
    6i − 3
    6i − 4
    6i
    6i − 11
    6i + 1

    View Slide

  23. Labeling Example C4
    P2
    S5
    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

  24. Cycle Pendant Trees
    Theorem
    Cn
    P2
    Sk
    is prime for all n ∈ N and 4 ≤ k ≤ 9.
    Conjecture
    Cn
    P2
    Sk
    is prime for all n, k ∈ N.

    View Slide

  25. Hairy Cycles
    An m-hairy cycle is a cycle Cn
    with m paths of length 2
    connected to each vertex of the cycle, denoted Cn
    mP2
    .
    We have found a prime labeling for 2, 3, 5, and 7-hairy
    cycles.
    Here is an example of a 2-hairy cycle, denoted C3
    2P2
    .
    a
    b c
    d
    e
    f
    h
    i
    j

    View Slide

  26. Cycles with 3 Hairs
    Theorem
    Cn
    3P2
    has a prime labeling.
    Labeling
    Let c1, c2, . . . cn
    denote the vertices of Cn
    , and let the pendant
    vertices adjacent to ci
    be denoted pj
    i
    , 1 ≤ j ≤ 3. The labeling
    function f : V → {1, 2, . . . 4n} is given by:
    f(ci) =
    1 i = 1
    4i − 1 i ≥ 2
    f(pj
    i
    ) =









    j + 1 i = 1, 1 ≤ j ≤ 3
    4i − 3 i ≥ 2, j = 1
    4i − 2 i ≥ 2, j = 2
    4i i ≥ 2, j = 3

    View Slide

  27. Example of a Cycle with Three Hairs
    1
    2
    3
    4
    15
    13
    14
    16
    7
    5
    6
    8
    11
    9
    10
    12

    View Slide

  28. 5-Hairy Cycle
    Theorem
    Cn
    5P2
    has a prime labeling for all n.
    Labeling
    Let c1, c2, . . . , cn
    denote the vertices of Cn
    , and let the pendant
    vertices adjacent ci
    be denoted pj
    i
    , 1 ≤ j ≤ 5. The labeling
    formula f : V → {1, 2, . . . , 6n} is given by:
    f(ci) =
    1, i = 1
    6(i − 1) + 5, i ≥ 2
    f(pj
    i
    ) =





    j + 1, 1 ≤ j ≤ 5, i = 1
    6(i − 1) + j, 1 ≤ j ≤ 4, i ≥ 2
    6(i − 1) + 6, j = 5, i ≥ 2

    View Slide

  29. Example of 5-Hairy Cycle
    1
    4
    5
    6
    2
    3
    23
    21
    22
    24
    19
    20
    17
    15
    16
    18
    13
    14
    11
    9
    10
    12
    7
    8

    View Slide

  30. Cycles with 7 Hairs
    Theorem
    Cn
    7P2
    has a prime labeling
    Labeling
    Let c1, c2, . . . cn
    denote the vertices of Cn
    , and let the pendant
    vertices adjacent to ci
    be denoted pi
    j
    , 1 ≤ j ≤ 7. The labeling
    formula f : V → {1, 2, . . . 8n} is given by:
    f(c1) = 1
    f(p1
    j
    ) = j + 1
    f(ci) =





    8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15)
    8i − 3 i ≡ 1, 4, 5, 7, 10, 13 (mod 15)
    8i − 1 i ≡ 0 (mod 15)
    f(pi
    j
    ) = {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}

    View Slide

  31. Example of a Cycle with 7 Hairs
    1
    2
    3
    4 5 6
    7
    8
    19
    17
    18
    20
    21
    22
    23
    24
    11
    9
    10
    12
    13
    14
    15
    16

    View Slide

  32. Shotgun Weed Graph
    Bertrand’s Postulate
    For every n > 1, there exists a prime p such that n ≤ p ≤ 2n.
    We can build a graph that is prime using this fact.
    1
    2
    13
    10
    9
    11
    14
    7
    12
    8
    5
    4
    3
    6

    View Slide

  33. And Now For Something Completely Different...
    Theorem
    If n is odd then Cn × Pi
    is not prime for all i > 1.
    Proof.
    Parity argument.
    1
    5
    4 3
    2
    7
    6
    10 9
    8

    View Slide

  34. Hastings Helms
    Theorem
    If n + 1 is prime then Cn × P2
    is prime.
    Labeling
    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

    View Slide

  35. Example of C6 × P2
    (Hastings Helms)
    1
    2 3
    4
    5
    6
    8
    9 10
    11
    12
    7

    View Slide

  36. 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.

    View Slide

  37. 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
    :

    View Slide

  38. Results
    Theorem
    All Sn × P4
    are prime.
    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

    View Slide

  39. Results
    Theorem
    All Sn × P6
    are prime.

    View Slide

  40. Results
    Labeling
    Let f : V → 1, 2, . . . , 6(n + 1) denote our labeling function. Let
    c1, c2, c3, c4, c5, c6
    form a path through the center of each Sn
    .
    Let
    f(c1) = 6, f(c2) = 1, f(c3) = 2, f(c4) = 3, f(c5) = 4,
    and f(c6) = p, the largest prime ≡ 1, 7, 13, 17, 23 (mod30). Then
    let vi,n
    be the vertex in the ith star and in the nth row. Then
    f(v1,1) = 7
    f(v2,1) = 8
    f(v3,1) = 9
    f(v4,1) = 10
    f(v5,1) = 11
    f(v6,1) = 12
    f(p) = 5
    f(vi,k ) = vi,k−1 + 6 for 1 ≤ i ≤ 6 and 2 ≤ k ≤ n

    View Slide

  41. Results
    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

    View Slide

  42. Conjecture
    Conjecture
    Sn × P12
    is prime.
    We have found a prime labeling that works for more than
    205,626 rows (2,467,524 vertices).
    Here is an example of 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

    View Slide

  43. Future Work
    Hairier Hairy Cycles?
    Larger Generalized Books?
    Larger Complete Ternary Trees?
    Lollipop Graphs?
    Unrooted Complete Binary Trees?
    ALL Unicyclic Graphs?????????

    View Slide

  44. Acknowledgements
    Jeff Rushall and Dana Ernst, advisors
    Department of Mathematics & Statistics at Northern Arizona
    University
    The Center for Undergraduate Research in Mathematics

    View Slide

  45. QUESTIONS?

    View Slide