New Results on Prime Vertex Labelings, I Levi Heath and Emily White Joint work with: Nathan Diefenderfer, Michael Hastings, Hannah Prawzinsky, Briahna Preston & Alyssa Whittemore SUnMaRC February 28, 2015

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.

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.

Connected and Unicyclic 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. Definition A unicyclic graph is a simple graph containing exactly one cycle. Here is a unicyclic graph that is NOT connected.

Pendants 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.

Infinite Families of Graphs P8 C12 S5

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 .

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

Prime Vertex Labeling 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.

Conjecture Conjecture (Seoud and Youssef, 1999) All unicyclic graphs have a prime vertex labeling.

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

New Infinite Families of Graphs Definition A cycle pendant star is a cycle with each cycle vertex adjacent to an identical star, denoted Cn P2 Sk . Definition A generalized cycle pendant star is a cycle with each cycle vertex adjacent to an identical star and each noncentral star vertex is a central vertex for an identical star, denoted Cn P2 Sk Sk Sk · · · Sk .

Example of C4 P2 S4

Prime Vertex Labelings of Infinite Families of Graphs Theorem For all n ∈ N, we have: The cycle pendant stars, Cn P2 Sk are prime for 3 ≤ k ≤ 8. The generalized cycle pendant star, Cn P2 S3 S3 has a prime vertex labeling. The cases for k ≥ 9 for cycle pendant stars and all other cases of generalized cycle pendant stars remain open.

Labeling Function of Cn P2 S3 Labeling Function Let c1, c2, . . . , cn denote the cycle labels, p1, p2, . . . , pn denote the vertices adjacent to the corresponding cycle vertices and the pendant vertices adjacent to 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

Prime Vertex Labeling of Cn P2 S3 1 3 4 2 5 11 13 14 12 15 6 7 9 8 10

Generalized Labeling of Cn P2 S4 6i − 5 6i − 1 6i − 4 6i − 3 6i − 2 6i 6i − 11 6i + 1

Prime Vertex Labeling of C4 P2 S4 1 5 2 3 4 6 19 23 20 21 22 24 7 11 8 9 10 12 13 17 14 15 16 18

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

EXCELing with Computer Aid

Prime Vertex Labeling of C4 P2 S7 1 5 2 3 4 6 7 8 9 10 13 11 12 14 15 16 17 18 19 23 20 21 22 24 25 26 27 28 31 29 30 32 33 34 35 36

Prime Vertex Labeling of Cn P2 S3 S3 1 2 5 9 11 3 4 6 7 8 10 12 13 14 15 16 19 23 25 17 18 20 21 22 24 26 27 28 29 32 31 35 41 30 33 34 36 37 38 39 40 42 43 44 47 51 53 45 46 48 49 50 52 54 55 56

Future Work Conjecture For all n, k ∈ N, Cn P2 Sk is prime. Conjecture For all n, k ∈ N, Cn P2 Sk Sk Sk · · · Sk is prime.

Acknowledgments SUnMaRC Organizers Center for Undergraduate Research in Mathematics Northern Arizona University Office of Undergraduate Research, NAU Research Advisors Dana Ernst and Jeff Rushall

Labeling Function of Cn P2 S3 S3 Note that Cn P2 S3 S3 contains 14n vertices. We will identify our vertices as follows. Let ci, 1 ≤ i ≤ n denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , let the non-cycle vertices adjacent to pi be denoted si,j for 1 ≤ j ≤ 3, and let the remaining vertices adjacent to each si,j be denoted li,j,k for 1 ≤ k ≤ 3. Our labeling function f : V → {1, 2, . . . , 14n} is best defined by first describing cycle and pendent vertex labels: f(ci) = 14i − 13, 1 ≤ i ≤ n f(pi) = 14i − 12, i ≡3 1, 2 14i − 10, i ≡3 0

Labeling Function of Cn P2 S3 S3 The remaining vertex labels are determined by the values of i, j and k as follows. If i ≡3 1, 2, then define f(si,j) =      14i − 9, j = 1 14i − 5, j = 2 14i − 3, j = 3 f(li,j,k ) =                                  14i − 11, j = 1, k = 1 14i − 10, j = 1, k = 2 14i − 8, j = 1, k = 3 14i − 7, j = 2, k = 1 14i − 6, j = 2, k = 2 14i − 4, j = 2, k = 3 14i − 2, j = 3, k = 1 14i − 1, j = 3, k = 2 14i, j = 3, k = 3

Labeling Function of Cn P2 S3 S3 If i ≡3 0, then define f(si,j) =      14i − 11, j = 1 14i − 7, j = 2 14i − 1, j = 3 f(li,j,k ) =                                  14i − 12, j = 1, k = 1 14i − 9, j = 1, k = 2 14i − 8, j = 1, k = 3 14i − 6, j = 2, k = 1 14i − 5, j = 2, k = 2 14i − 4, j = 2, k = 3 14i − 3, j = 3, k = 1 14i − 2, j = 3, k = 2 14i, j = 3, k = 3

Labeling Function of Cn P2 S4 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

Labeling Function of Cn P2 S5 Let ci , 1 ≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 5, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 7n} is given by: f(ci) = 7i − 6, 1 ≤ i ≤ n f(pi) =                7i − 2, i ≡6 1, 3 7i − 3, i ≡6 2, 4 7i − 4, i ≡6 5 7i − 5, i ≡6 0, i ≡30 0 7i − 1, i ≡30 0

Labeling Function of Cn P2 S5 f(oi1) = 7i − 5, i ≡6 0 or i ≡30 0 7i − 4, i ≡6 0, i ≡30 0 f(oi2) = 7i − 4, i ≡6 0, 5 or i ≡30 0 7i − 3, i ≡6 5 or i ≡6 0, i ≡30 0 f(oi3) = 7i − 3, i ≡6 0, 2, 4, 5 or i ≡30 0 7i − 2, i ≡6 1, 3 f(oi4) = 7i − 2, i ≡30 0 7i − 1, i ≡30 0 f(oi5) = 7i, 1 ≤ i ≤ n

Labeling Function of Cn P2 S6 Let ci , 1 ≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 6, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 8n} is given by: f(ci) = 8i − 7, 1 ≤ i ≤ n f(pi) =      8i − 3, i ≡3 0 8i − 5, i ≡3 0, i ≡15 0 8i − 1, i ≡15 0

Labeling Function of Cn P2 S6 f(oi1) = 8i − 6, 1 ≤ i ≤ n f(oi2) = 8i − 5, i ≡3 0 or i ≡15 0 8i − 3, i ≡3 0, i ≡15 0 f(oi3) = 8i − 4, 1 ≤ i ≤ n f(oi4) = 8i − 3, i ≡15 0 8i − 1, i ≡15 0 f(oi5) = 8i − 2, 1 ≤ i ≤ n f(oi6) = 8i, 1 ≤ i ≤ n

Labeling Function of Cn P2 S7 Let ci , 1 ≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 7, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 9n} is given by: f(ci) = 9i − 8, 1 ≤ i ≤ n f(pi) =          9i − 4, i ≡2 1 9i − 5, i ≡2 0, i ≡10 0 9i − 7i ≡10 0, i ≡70 0 9i − 1, i ≡70 0

Labeling Function of Cn P2 S7 f(oi1) = 9i − 7, i ≡10 0 or i ≡70 0 9i − 5, i ≡10 0, i ≡70 0 f(oi2) = 9i − 6, 1 ≤ i ≤ n f(oi3) = 9i − 5, i ≡2 1 or i ≡70 0 9i − 4, i ≡2 0, i ≡70 0 f(oi4) = 9i − 4, i ≡70 0 9i − 1, i ≡70 0 f(oi5) = 9i − 3, 1 ≤ i ≤ n f(oi6) = 9i − 2, 1 ≤ i ≤ n f(oi7) = 9i, 1 ≤ i ≤ n

Labeling Function of Cn P2 S8 Let ci , 1 ≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 8, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 10n} is given by: f(ci) = 10i − 9, 1 ≤ i ≤ n f(pi) =      10i − 3, i ≡3 0 10i − 7, i ≡3 0, i ≡21 0 10i − 1, i ≡21 0

Labeling Function of Cn P2 S8 f(oi1) = 10i − 8, 1 ≤ i ≤ n f(oi2) = 10i − 7, i ≡21 0 or i ≡3 1, 2 10i − 3, i ≡3 0, i ≡21 0 f(oi3) = 10i − 6, 1 ≤ i ≤ n f(oi4) = 10i − 5, 1 ≤ i ≤ n f(oi5) = 10i − 4, 1 ≤ i ≤ n f(oi6) = 10i − 3, i ≡21 0 10i − 1, i ≡21 0 f(oi7) = 10i − 2, 1 ≤ i ≤ n f(oi8) = 10i, 1 ≤ i ≤ n