Prime Vertex Labelings of Graphs

This poster was presented by my undergraduate research students Hannah Prawzinsky and Emily White on January 25, 2015 at the Nebraska Conference for Undergraduate Women in Mathematics at the University of Nebraska-Lincoln.

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

Transcript

1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Prime Vertex Labelings Hannah Prawzinsky & Emily White, Directed by Dana C. Ernst & Jeff Rushall Department of Mathematics & Statistics, Northern Arizona University 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. Examples Definition A simple graph is a graph that contains neither “loops” nor multiple edges between vertices. Example Here is an example of a graph that is not simple. Definition 1. The path Pn consists of n vertices and n − 1 edges such that 2 vertices have degree 1 and n − 2 vertices have degree 2. Here is P8 . 2. The cycle Cn consists of n vertices and n edges such that each vertex has degree 2. Here is C12 . 3. The star Sn consists of one vertex of degree n and n vertices of degree one. Here is S5 . Definition G1 ⋆ G2 is the graph that results from “selectively gluing” copies of G2 to specified vertices of G1 . Here is C4 ⋆ P2 ⋆ S4 . Definition A graph labeling is an “assignment” of integers (possibly satisfying some con- ditions) to the vertices, edges, or both. Formal graph labelings are functions. Here is a vertex labeling, an edge labeling, and a vertex-edge labeling, respec- tively, for C4 . 2 3 2 3 1 4 1 4 1 2 3 4 1 2 3 4 Definition A graph with n vertices has a prime vertex labeling if its vertices can be labeled with the integers 1, 2, 3, . . . , n such that every pair of adjacent vertices (i.e., vertices that share an edge) have labels that are relatively prime. Theorem 1. All paths have a prime vertex labeling. 1 2 3 4 5 6 7 8 2. All cycles have a prime vertex labeling. 1 2 3 4 5 6 7 8 9 10 11 12 3. All starts have a prime vertex labeling. 1 2 3 4 5 6 Conjecture (Seoud) All unicyclic graphs (i.e., graphs containing exactly one cycle) have a prime vertex labeling. Theorem (Prawzinksky & White et al.) 1. Every Cn ⋆ P2 ⋆ S3 has a prime vertex labeling. 2. Every Cn ⋆ P2 ⋆ S3 ⋆ S3 has a prime vertex labeling. Labeling for Part 1 of Theorem Let c1, c2, . . . , cn denote the cycle labels, let p1, p2, . . . , pn denote the vertices adjacent to the corresponding cycle vertices and let the pendant vertices off of 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 Example Here is an example of the labeling for part 1 of our theorem. 1 3 4 2 5 11 13 14 12 15 6 7 9 8 10 Additional New Results We have also discovered prime vertex labelings for the following graphs: 1. Double and triple-tailed tadpoles 2. Cn ⋆ P2 ⋆ Sk for 3 ≤ k ≤ 8 3. Cn ⋆ S3 , Cn ⋆ S5 , Cn ⋆ S7 4. Cn ⋆ 3P2 , Cn ⋆ 5P2 , Cn ⋆ 7P2 In addition, we have obtained results for a few families of non-unicyclic graphs. Email: Hannah [hpp3@nau.edu], Emily [ekw49@nau.edu] Typeset using L A TEX, TikZ, and beamerposter