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).

Transcript

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

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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)
   

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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.
   

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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.
   

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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.
   

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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
   .
   

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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
   .
   

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8. Example
   Here is C3
   S5
   .
   

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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
   

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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
    

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

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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
    

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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
    

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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.
    

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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
    

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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.
    

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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
    

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

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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
    .
    

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20. Example C4
    P2
    S5
    

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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
    

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

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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
    

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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.
    

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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
    

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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
    

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

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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
    

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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
    

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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)}
    

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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
    

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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
    

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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
    

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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
    

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

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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.
    

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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
    :
    

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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
    

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39. Results
    Theorem
    All Sn × P6
    are prime.
    

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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
    

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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
    

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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
    

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

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

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45. QUESTIONS?
    

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