Prime Vertex Labelings of Graphs Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, Briahna Preston, Emily White, & Alyssa Whittemore FAMUS December 5, 2014

Why are we here? CURM (Center for Undergraduate Research in Math) Mini-grant MAT 485 Conferences NCUWM - 1/24 (Lincoln, NE) SUnMaRC - 2/27 (El Paso, TX) CURM Conference - 3/20 (Provo, UT)

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 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. Here is a graph that is NOT connected.

Cycles Definition A cycle is a connected path consisting of a single “circuit,” and hence has equal numbers of vertices and edges. We let Cn denote an n-vertex cycle. Below is an image of C12 .

Unicyclic Graphs Definition A unicyclic graph (simple and connected) is a graph containing exactly one cycle. Definition G1 G2 is the graph that results from “selectively gluing” copies of G2 to some vertices of G1 .

Example Here is C3 S5 .

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 Labelings Definition A graph with n vertices has a prime vertex labeling if its vertices are labeled with the integers 1, 2, 3, . . . , n and every pair of 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 Labeling Use integers 1–12 Adjacent vertices must have relatively prime labels No two vertices can have identical labels

Conjecture Conjecture (Seoud) All unicyclic graphs have a prime vertex labeling.

Cycles and Prime Vertex Labelings Theorem All cycles have a prime vertex labeling. 1 2 3 4 5 6 7 8 9 10 11 12

More Examples Here are examples of infinite families of graphs with prime vertex labelings. 1 2 3 4 5 6 7 8 16 15 14 13 12 11 10 9 1 2 3 4 5 6 7 8

More Examples (continued) 1 2 3 4 5 6 1 11 9 7 5 3 2 12 10 8 6 4 The following presenters will discuss some of their original findings for specific families of graphs.

Ternary Trees Definition A complete ternary tree is a directed rooted tree with every internal vertex having 3 children. Here are examples of 1 and 2-level complete ternary trees: a b c d a b c d b1 b2 b3 c1 c2 c3 d1 d2 d3

New Results Theorem 1 Every Cn P2 S4 has a prime vertex labeling. Theorem 2 Every Cn P2 S4 S4 has a prime vertex labeling.

Theorem 1 Labeling 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 of Theorem 1 Labeling 1 3 4 2 5 11 13 14 12 15 6 7 9 8 10

Cycle Pendant Stars Definition A star is a graph with one vertex adjacent to all other vertices, each of which has degree one. Definition A cycle pendant star is a cycle with each cycle vertex adjacent to an identical star, denoted Cn P2 Sk .

Example C4 P2 S5

Cn P2 S5 Theorem Cn P2 S5 is prime for all n. Labeling 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

General Labeling 6i − 5 6i − 1 6i − 2 6i − 3 6i − 4 6i 6i − 11 6i + 1

Labeling Example C4 P2 S5 1 5 2 3 4 6 19 23 20 21 22 24 7 11 8 9 10 12 13 17 14 15 16 18

Cycle Pendant Trees Theorem Cn P2 Sk is prime for all n ∈ N and 4 ≤ k ≤ 9. Conjecture Cn P2 Sk is prime for all n, k ∈ N.

Hairy Cycles An m-hairy cycle is a cycle Cn with m paths of length 2 connected to each vertex of the cycle, denoted Cn mP2 . We have found a prime labeling for 2, 3, 5, and 7-hairy cycles. Here is an example of a 2-hairy cycle, denoted C3 2P2 . a b c d e f h i j

Cycles with 3 Hairs Theorem Cn 3P2 has a prime labeling. Labeling Let c1, c2, . . . cn denote the vertices of Cn , and let the pendant vertices adjacent to ci be denoted pj i , 1 ≤ j ≤ 3. The labeling function f : V → {1, 2, . . . 4n} is given by: f(ci) = 1 i = 1 4i − 1 i ≥ 2 f(pj i ) =          j + 1 i = 1, 1 ≤ j ≤ 3 4i − 3 i ≥ 2, j = 1 4i − 2 i ≥ 2, j = 2 4i i ≥ 2, j = 3

Example of a Cycle with Three Hairs 1 2 3 4 15 13 14 16 7 5 6 8 11 9 10 12

5-Hairy Cycle Theorem Cn 5P2 has a prime labeling for all n. Labeling Let c1, c2, . . . , cn denote the vertices of Cn , and let the pendant vertices adjacent ci be denoted pj i , 1 ≤ j ≤ 5. The labeling formula f : V → {1, 2, . . . , 6n} is given by: f(ci) = 1, i = 1 6(i − 1) + 5, i ≥ 2 f(pj i ) =      j + 1, 1 ≤ j ≤ 5, i = 1 6(i − 1) + j, 1 ≤ j ≤ 4, i ≥ 2 6(i − 1) + 6, j = 5, i ≥ 2

Example of 5-Hairy Cycle 1 4 5 6 2 3 23 21 22 24 19 20 17 15 16 18 13 14 11 9 10 12 7 8

Cycles with 7 Hairs Theorem Cn 7P2 has a prime labeling Labeling Let c1, c2, . . . cn denote the vertices of Cn , and let the pendant vertices adjacent to ci be denoted pi j , 1 ≤ j ≤ 7. The labeling formula f : V → {1, 2, . . . 8n} is given by: f(c1) = 1 f(p1 j ) = j + 1 f(ci) =      8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15) 8i − 3 i ≡ 1, 4, 5, 7, 10, 13 (mod 15) 8i − 1 i ≡ 0 (mod 15) f(pi j ) = {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}

Example of a Cycle with 7 Hairs 1 2 3 4 5 6 7 8 19 17 18 20 21 22 23 24 11 9 10 12 13 14 15 16

Shotgun Weed Graph Bertrand’s Postulate For every n > 1, there exists a prime p such that n ≤ p ≤ 2n. We can build a graph that is prime using this fact. 1 2 13 10 9 11 14 7 12 8 5 4 3 6

And Now For Something Completely Different... Theorem If n is odd then Cn × Pi is not prime for all i > 1. Proof. Parity argument. 1 5 4 3 2 7 6 10 9 8

Hastings Helms Theorem If n + 1 is prime then Cn × P2 is prime. Labeling Let c1 1 , c1 2 , . . . , c1 n denote the vertices on the inner cycle, and c2 1 , c2 2 , ..., c2 n be the vertices on the outer cycle. The labeling formula f : V → {1, 2, . . . , 2n} is given by: f(cj i ) =      i, i = 1, 2, . . . , n, j = 1 i + n + 1, i = 1, 2, . . . , n − 1, j = 2 n + 1, i = n, j = 2

Example of C6 × P2 (Hastings Helms) 1 2 3 4 5 6 8 9 10 11 12 7

Books Definition A book is the graph Sn × P2 , where Sn is the star with n pendant vertices and P2 is the path with 2 vertices. Here is a picture of S4 × P2 : 3 5 7 9 4 6 8 10 2 1 It is known that all books have a prime labeling.

Book Generalizations Definition A generalized book is a graph of the form Sn × Pm , which looks like m − 1 books glued together. Here is a picture of S5 × P4 :

Results Theorem All Sn × P4 are prime. Here is an example of S5 × P4 : 24 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 3 23 2 1 4

Results Theorem All Sn × P6 are prime.

Results Labeling Let f : V → 1, 2, . . . , 6(n + 1) denote our labeling function. Let c1, c2, c3, c4, c5, c6 form a path through the center of each Sn . Let f(c1) = 6, f(c2) = 1, f(c3) = 2, f(c4) = 3, f(c5) = 4, and f(c6) = p, the largest prime ≡ 1, 7, 13, 17, 23 (mod30). Then let vi,n be the vertex in the ith star and in the nth row. Then f(v1,1) = 7 f(v2,1) = 8 f(v3,1) = 9 f(v4,1) = 10 f(v5,1) = 11 f(v6,1) = 12 f(p) = 5 f(vi,k ) = vi,k−1 + 6 for 1 ≤ i ≤ 6 and 2 ≤ k ≤ n

Results Here is an example of the prime labeling for Sn × P6 , in particular, S4 × P6 : 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 5 24 25 26 27 28 29 30 6 1 2 3 4 23

Conjecture Conjecture Sn × P12 is prime. We have found a prime labeling that works for more than 205,626 rows (2,467,524 vertices). Here is an example of S3 × P12 : 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 10 33 34 35 36 37 38 39 40 11 42 5 44 45 46 7 48 2 3 4 1 6 43 8 9 32 47 12 41

Future Work Hairier Hairy Cycles? Larger Generalized Books? Larger Complete Ternary Trees? Lollipop Graphs? Unrooted Complete Binary Trees? ALL Unicyclic Graphs?????????

Acknowledgements Jeff Rushall and Dana Ernst, advisors Department of Mathematics & Statistics at Northern Arizona University The Center for Undergraduate Research in Mathematics

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