. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 [

[email protected]], Emily [

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