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).

Transcript

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