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