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New Prime Vertex Labelings

Dana Ernst
April 24, 2015

New Prime Vertex Labelings

This poster was presented by my undergraduate research students Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, and Alyssa Whittemore (Northern Arizona University) on April 25, 2015 at the 2015 NAU Undergraduate Symposium at Northern Arizona University in Flagstaff, AZ.

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

Dana Ernst

April 24, 2015
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  1. New Prime Vertex Labelings
    N. Diefenderfer, M. Hastings, L. Heath, H. Prawzinsky, B. Preston, E. White, & A. Whittemore
    Department of Mathematics & Statistics, Northern Arizona University
    Simple Graph
    Definition: A simple graph G(V, E) is a set V of vertices and a set E of
    edges connecting some (possibly empty) subset of those vertices such that
    G(V, E) does not contain loops or multiple edges. The degree of a vertex is
    the number of edges “touching” that vertex.
    Below is a graph that is not simple.
    Families of Graphs
    Definition: For n ≥ 2, a path, denoted Pn
    , is a connected graph that
    consists of n vertices and n − 1 edges such that 2 vertices have degree 1 and
    n − 2 vertices have degree 2.
    Definition: For n ≥ 3, a cycle, denoted Cn
    , a connected graph consisting
    of n vertices, each of degree 2.
    Definition: A star, denoted Sn
    , consists of one vertex of degree n and n
    vertices of degree one.
    P8
    C12
    S5
    Graph Labeling
    Definition: A graph labeling is an “assignment” of integers (possibly sat-
    isfying some conditions) to the vertices, edges, or both.
    Definition: A graph with n vertices has a prime vertex labeling, or is said
    to be coprime, 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.
    Gluing Function
    Definition: G1
    G2
    is the graph that results from“selectively gluing” copies
    of G2
    to some vertices of G1
    .
    Here is C3
    P2
    S3
    .
    Cycle Pendant Stars
    Definition: A cycle pendant star is the graph of the form Cn
    P2
    Sk
    and
    a generalized cycle pendant star is the graph of the form Cn
    P2
    Sk
    Sk
    Sk
    · · · Sk
    .
    Theorem: All cycle pendant stars of the form Cn
    P2
    Sk
    for 3 ≤ k ≤ 8,
    and all generalized cycle pendant stars of the form Cn
    P2
    S3
    S3
    have a
    prime vertex labeling.
    Below is a prime vertex labeling of C5
    P2
    S6
    .
    1
    5
    2
    3 4 6 7
    8
    9
    13
    10
    11
    12
    14
    15
    16
    17
    19 18
    20
    21
    22
    23
    24
    25
    29
    26
    27
    28
    30
    31
    32
    33
    37
    34
    35
    36
    38
    39
    40
    Hairy Cycles
    Definition: An m-hairy cycle is a graph of the form Cn
    Sm
    .
    Theorem: All 3-hairy, 5-hairy and 7-hairy cycles are coprime.
    Below is a prime vertex labeling of C4
    S7
    .
    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
    Generalized Books
    Definition: A book is a graph of the form Sn
    × P2
    , and a generalized book
    is a graph of the form Sn
    × Pm
    .
    Theorem: All Sn
    × Pm
    are coprime for 3 ≤ m ≤ 7.
    Below is a prime vertex labeling of S8
    × P6
    .
    1 2 3 4 5 6
    12 11 10 9 8 7
    18 17 16 15 14 13
    20 21 22 23 24 19
    30 29 28 27 26 25
    36 35 34 33 32 31
    42 41 40 39 38 37
    48 47 46 45 44 43
    50 51 52 53 54 49
    Prisms
    Definition: A prism is a graph of the form Cn
    × P2
    .
    Theorem: If n − 1 is a prime number and n ≥ 4, then Cn
    × P2
    is coprime.
    Below is a prime vertex labeling of C6
    × P2
    .
    5
    2
    3
    4 1
    12
    6
    7
    8
    9 10
    11
    Cycle Chains
    Definition: A cycle chain, denoted Cm
    n
    , is a graph consisting of m distinct
    n-cycles symmetrically adjoined by a single vertex on each cycle.
    Definition: A Mersenne prime is a prime number of the form Mn
    = 2n −1.
    Theorem: All Cm
    8
    , Cm
    6
    , Cm
    4
    and Cm
    n
    , and where n = 2k and 2k − 1 is a
    Mersenne prime have prime vertex labelings.
    Here is a prime vertex labeling of C5
    8
    .
    1
    2
    3
    4
    5
    6
    7
    8
    15
    11
    10
    9
    1
    12
    13
    14
    19
    18
    17
    16
    15
    22
    21
    20
    29
    25
    24
    23
    19
    26
    27
    28
    33
    32
    31
    30
    29
    36
    35
    34
    Fibonacci Chains
    Definition: A Fibonacci chain, denoted Cm
    F
    , is a graph consisting of m
    cycles adjoined by a single vertex on each cycle, where the first cycle contains
    3 vertices and the ith cycle contains Fi+1
    + 1 vertices.
    Theorem: All Cm
    F
    are coprime.
    Below is an example of a prime vertex labeling of C5
    F
    .
    1
    2
    4
    3 5
    6 7
    10
    9
    8
    12
    11
    13
    14
    15
    16
    17
    18
    19
    20
    21
    References
    1. Gallian. A dynamic survey of graph labeling. Electron. J. Comb.,
    Volume 17, 2014.
    2. Diefenderfer, Hastings, Heath, Prawzinsky, Preston, White, Whittemore.
    Prime Vertex Labelings of Families of Unicyclic Graphs.
    Rose-Hulman Undergraduate Mathematics Journal (to appear).
    3. Diefenderfer, Ernst, Hastings, Heath, Prawzinsky, Preston, Rushall,
    White, Whittemore. Prime Vertex Labelings of Several Families of
    Graphs (submitted).
    Directed by Dana C. Ernst & Jeff Rushall Typeset using L
    A
    TEX, TikZ, and beamerposter

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