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Prime Vertex Labelings of Graphs

Dana Ernst
January 24, 2015

Prime Vertex Labelings of Graphs

This talk was given by my undergraduate research students Briahna Preston and Alyssa Whittemore on January 25, 2015 at the Nebraska Conference for Undergraduate Women in Mathematics at the University of Nebraska-Lincoln.

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

Dana Ernst

January 24, 2015
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  1. Prime Vertex Labelings of Graphs
    Briahna Preston & Alyssa Whittemore (joint work with Nathan
    Diefenderfer, Michael Hastings, Levi Heath, Hannah
    Prawzinsky, & Emily White)
    NCUWM
    January 24, 2015
    Funded by a CURM Mini-grant

    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. 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.

    View Slide

  3. Examples of Graphs
    Definition
    A path, denoted Pn
    , consists of n vertices and n − 1 edges such
    that 2 vertices have degree 1 and n − 2 vertices have degree 2.
    Definition
    A cycle, denoted Cn
    , is a closed “loop” with n vertices and n
    edges with every vertex of degree 2.
    Definition
    A star, denoted Sn
    , consists of one vertex of degree n and n
    vertices of degree one.

    View Slide

  4. Constructing New Graphs From Old Graphs
    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

  5. 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 such that no two
    vertices have the same label 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

  6. Families with Prime Vertex Labelings
    1
    2
    3
    4
    5 6
    1
    11
    9
    7
    5 3
    2
    12
    10
    8
    6 4
    Among the infinite families known to have prime vertex
    labelings are stars, pendant graphs, cycles, paths, ladders,
    tadpoles, ...

    View Slide

  7. Families without Prime Vertex Labelings
    3
    2
    6 5
    4
    1
    Among the infinite families known to not have prime vertex
    labelings are Wn
    if n odd, Kn
    if n ≥ 4, K2 + Cn
    if n ≥ 4, ...

    View Slide

  8. Our Research Project
    Our research project involves finding new infinite families of
    graphs with prime vertex labelings. Part of our focus has been
    on the following conjecture:
    Conjecture (Seoud, 1990)
    All unicyclic graphs (graphs containing exactly one cycle)
    have a prime vertex labeling.

    View Slide

  9. New Results
    One of the simplest types of unicyclic graphs is a tadpole,
    which is a cycle with one path attached. It is trivial to show that
    all tadpoles have a prime vertex labeling.
    It is somewhat less trivial to prove the following:
    Theorem 1 (Preston & Whittemore)
    All double- and triple-tailed tadpoles have prime vertex
    labelings.
    9 2
    3
    4
    5
    6
    7
    8
    10
    11
    12
    13
    1 17
    16
    15
    14

    View Slide

  10. New Results
    Theorem 2 (Preston & Whittemore et al.)
    The following infinite families of unicyclic graphs have prime
    vertex labelings:
    Cn
    P2
    S3
    Cn
    P2
    S3
    S3
    Cn
    P2
    Sk
    for 3 ≤ k ≤ 8
    Cn
    S3
    Cn
    S5
    Cn
    S7

    View Slide

  11. New Results
    Here is an example for Cn
    S5
    , specifically C4
    S5
    :
    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

  12. New Results
    Theorem 3 (Preston & Whittemore et al.)
    The following infinite families of graphs (non-unicyclic) have
    prime vertex labelings:
    Sn × P4
    Sn × P6
    Cn × P2
    if n + 1 is a prime number

    View Slide

  13. New Results
    Here is an example of the prime labeling for Sn × P6
    , specifically
    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

  14. New Results
    Theorem 4 (Preston & Whittemore et al.)
    The family Cn × Pi
    if n is odd for all i > 1 does not have a prime
    vertex labeling.
    1
    5
    4 3
    2
    7
    6
    10 9
    8

    View Slide

  15. Future Work
    Conjectures
    Here are some infinite families we hope to show have prime
    vertex labelings:
    Cn
    P2
    Sk
    for all n ≥ 3, k ≥ 3
    Sn × P12
    All unicyclic graphs (Seoud’s Conjecture)

    View Slide

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

    View Slide

  17. QUESTIONS?

    View Slide

  18. Proof
    An odd-cycle double-tailed tadpole with n vertices is labeled
    with the following labeling function:
    f(vi) =









    i + 1, for i ≤ k
    i + 2, for k < i ≤ n − 2
    k + 1, for i = n
    1, for i = n − 1
    Note that v1
    must be the vertex adjacent to an intersection
    point on the even arc length of the cycle and vk
    is the other
    intersection point.

    View Slide

  19. 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

  20. 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

  21. 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

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

    View Slide

  23. 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

  24. Example C4
    P2
    S5

    View Slide

  25. 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

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

    View Slide

  27. 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

  28. 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

  29. 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

  30. 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 pi
    j
    , 1 ≤ j ≤ 3. The labeling
    function f : V → {1, 2, . . . 4n} is given by:
    f(ci) =
    1 i = 1
    4i − 1 i ≥ 2
    f(pi
    j
    ) =









    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

  31. 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

  32. 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 pi
    j
    , 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(pi
    j
    ) =





    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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

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

    View Slide

  40. 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

  41. 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

  42. 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

  43. Results
    Theorem
    All Sn × P6
    are prime.

    View Slide

  44. 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

  45. 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

  46. 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

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

    View Slide

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

    View Slide

  49. QUESTIONS?

    View Slide