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).

Transcript

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