Unicyclic Graphs with Prime Vertex Labelings, II

This talk was given by my undergraduate research students Nathan Diefenderfer and Briahna Preston on March 20, 2015 at the 2015 MAA/CURM Spring Conference at Brigham Young University.

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

Transcript

1. Unicyclic Graphs with Prime Vertex
   Labelings, II
   Nathan Diefenderfer and Briahna Preston
   Joint work with: Michael Hastings, Levi Heath, Hannah
   Prawzinsky, Emily White & Alyssa Whittemore
   2015 MAA/CURM Spring Conference
   March 20, 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. 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.
   

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3. Graph Theory Terminology
   Definition
   Two vertices are considered adjacent if there is an edge
   between them.
   Definition
   The degree of a vertex is the number of edges having that
   vertex as an endpoint.
   Definition
   A subgraph H of a graph G is a graph whose vertex set is a
   subset of that of G, and whose adjacency relation is a subset of
   that of G restricted to this subset.
   

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4. Infinite Families of Graphs
   P8
   C12
   S5
   

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5. Prime Vertex Labelings
   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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6. 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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7. Unicyclic Graphs
   Definition
   A graph is unicyclic if it contains only one cycle as a subgraph.
   Conjecture (Seoud and Youssef, 1999)
   All unicyclic graphs have a prime labeling.
   5
   1
   2
   3 4
   9
   8 10
   7
   6
   

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8. Tadpoles
   Definition
   A tadpole is a unicyclic graph composed of a cycle and a
   single path starting at what we call an intersection point, a
   vertex in the cycle.
   Definition
   An arc length is the number of cycle edges between two
   intersection points. So an even or odd arc length refers to the
   parity of the number of edges in the arc length.
   

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9. Double-Tailed Tadpoles
   Theorem
   All double-tailed tadpoles have a prime labeling.
   Labeling - Case 1
   The most simple case, a double-tailed tadpole with n vertices
   and one intersection point vk
   , is labeled with the following
   labeling function.
   f(vi) =
   
   
   
   
   
   i + 1, for i < k
   1, for i = k
   i, for i > k
   

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10. A Labeled 6-Cycle Double-Tailed Tadpole
    3
    4
    5
    6
    1 2
    13
    12
    11
    7
    8
    9
    10
    

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11. Even-Cycle Double-Tailed Tadpole
    Labeling - Case 2
    An even-cycle double-tailed tadpole with n vertices and two
    distinct intersection points is labeled with the following labeling
    function:
    f(vi) =
    i + 1, for i < n
    1, for i = n
    Note that v1
    must be the vertex adjacent to an intersection
    point.
    

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12. A Labeled 8-Cycle Double-Tailed Tadpole
    4
    5
    6
    7
    8
    9
    2
    3
    1 15 14 13
    10
    11
    12
    

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13. Odd-Cycle Double-Tailed Tadpole
    Labeling - Case 3
    An odd-cycle double-tailed tadpole with n vertices and two
    distinct intersection points 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.
    

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14. A Labeled 7-Cycle Double-Tailed Tadpole
    5
    6
    7
    8
    9
    2
    3
    4 1 14
    10
    11
    12
    13
    

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15. Triple-Tailed Tadpoles
    Theorem
    All odd-cycle triple-tailed tadpoles have a prime labeling.
    Labeling - Case 1
    An odd-cycle triple tailed tadpole with n vertices, three distinct
    intersection points, and three odd tails is labeled with the
    following function:
    f(vi) =
    
    
    
    
    
    
    
    
    
    i + 1, for 1 ≤ i ≤ 2
    i + 2, for 3 ≤ i < n − 2
    4, for i = j
    1, for i = k
    

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16. A Labeled 9-cycle Triple-Tailed Tadpole
    7
    8
    9
    10
    11
    2
    3
    5
    6
    1 20 19
    12
    13
    14
    15
    16
    4 17 18
    

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17. Hairy Cycles
    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.
    Definition
    An m-hairy cycle, denoted Cn
    Sm
    is a cycle with m pendants
    adjacent to each cycle vertex.
    

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18. Basic Approach
    For an m-hairy cycle:
    • N is split into sets of m + 1 consecutive integers.
    • Each set is assigned to a cycle vertex and its pendants.
    • A number relatively prime to the rest is assigned to the
    cycle vertex.
    • Odd m values were used so the number of evens and odds
    in each set was consistent.
    

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19. 3-Hairy Cycle
    Theorem
    For all n ≥ 3, Cn
    S3
    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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20. A Labeled 3-Hairy Cycle
    1
    2
    3
    4
    15
    13
    14
    16
    7
    5
    6
    8
    11
    9
    10
    12
    

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21. 5-Hairy Cycle
    Theorem
    For all n ≥ 3, Cn
    S5
    has a prime labeling.
    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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22. A Labeled 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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23. 7-Hairy Cycle
    Theorem
    For all n ≥ 3, Cn
    S7
    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 ≤ 7. The labeling
    formula f : V → {1, 2, . . . 8n} is given by:
    f(c1) = 1
    f(pj
    1
    ) = j + 1
    f(ci) =
    
    
    
    
    
    8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15)
    8i − 3 i ≡ 4, 5, 7, 10, 13 (mod 15)
    8i − 1 i ≡ 0, 1 (mod 15)
    f(pj
    i
    ) ∈ {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}
    

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24. A Labeled 7-Hairy Cycle
    1
    2
    3
    4 5 6
    7
    8
    19
    17
    18
    20
    21
    22
    23
    24
    11
    9
    10
    12
    13
    14
    15
    16
    29
    25
    26
    27
    28
    30
    31
    32
    

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25. Future Work
    This technique works well for odd m values, so extending this
    method to evens is an obvious goal.
    However, this method will fail for m ≥ 16, due to a result by
    S.S. Pillai, who found that in a string of 17 or more consecutive
    integers, there will not always be an integer in the set relatively
    prime to the rest.
    

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26. Bertrand’s Postulate
    Prior to this, labelings were constructed for graphs. However, for
    this graph, a labeling scheme was devised first.
    Bertrand’s Postulate
    For every n > 1, there exists a prime p such that n < p < 2n.
    Unlike hairy cycles, this constructed graph will not have a
    consistent number of pendants per cycle vertex. Instead, the
    number will grow to allow us to use Bertrand’s postulate to find
    primes.
    

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27. Bertrand Weed Graph
    Earlier, the natural numbers were split up into sets of a
    consistent size. For this graph, each set of integers needs to be
    twice as large as the previous set.
    Due to Bertrand’s postulate, there is guaranteed to be a prime
    in each set. Assign the largest prime in each set as the label to
    the cycle vertex, then label the pendant vertices with the rest
    of the consecutive integers.
    

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28. A Labeled Bertrand Weed Graph
    1
    2
    13
    10
    9
    11
    14
    7
    12
    8
    5
    4
    3
    6
    

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29. Acknowledgments
    Center for Undergraduate Research in Mathematics
    Northern Arizona University
    Research Advisors Dana Ernst and Jeff Rushall
    

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