Prime Vertex Labelings of Graphs

This talk was given by my undergraduate research students Briahna Preston and Alyssa Whittemore 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. Prime Vertex Labelings of Graphs Briahna Preston & Alyssa Whittemore (joint work with Nathan Diefenderfer, Michael Hastings, Levi Heath, Hannah Prawzinsky, & Emily White) NCUWM January 24, 2015 Funded by a CURM Mini-grant
2. 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. A simple graph is a graph that contains neither “loops” nor multiple edges between vertices. A connected graph is a graph in which there exists a “path” between every pair of vertices. For the remainder of the presentation, all graphs are assumed to be simple and connected.
3. Examples of Graphs Definition A path, denoted Pn , consists of n vertices and n − 1 edges such that 2 vertices have degree 1 and n − 2 vertices have degree 2. Definition A cycle, denoted Cn , is a closed “loop” with n vertices and n edges with every vertex of degree 2. Definition A star, denoted Sn , consists of one vertex of degree n and n vertices of degree one.
4. Constructing New Graphs From Old Graphs Definition G1 G2 is the graph that results from “selectively gluing” copies of G2 to some vertices of G1 . Here is C3 P2 S3 .
5. 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 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. 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
6. Families with Prime Vertex Labelings 1 2 3 4 5 6 1 11 9 7 5 3 2 12 10 8 6 4 Among the infinite families known to have prime vertex labelings are stars, pendant graphs, cycles, paths, ladders, tadpoles, ...
7. Families without Prime Vertex Labelings 3 2 6 5 4 1 Among the infinite families known to not have prime vertex labelings are Wn if n odd, Kn if n ≥ 4, K2 + Cn if n ≥ 4, ...
8. Our Research Project Our research project involves finding new infinite families of graphs with prime vertex labelings. Part of our focus has been on the following conjecture: Conjecture (Seoud, 1990) All unicyclic graphs (graphs containing exactly one cycle) have a prime vertex labeling.
9. New Results One of the simplest types of unicyclic graphs is a tadpole, which is a cycle with one path attached. It is trivial to show that all tadpoles have a prime vertex labeling. It is somewhat less trivial to prove the following: Theorem 1 (Preston & Whittemore) All double- and triple-tailed tadpoles have prime vertex labelings. 9 2 3 4 5 6 7 8 10 11 12 13 1 17 16 15 14
10. New Results Theorem 2 (Preston & Whittemore et al.) The following infinite families of unicyclic graphs have prime vertex labelings: Cn P2 S3 Cn P2 S3 S3 Cn P2 Sk for 3 ≤ k ≤ 8 Cn S3 Cn S5 Cn S7
11. New Results Here is an example for Cn S5 , specifically C4 S5 : 1 4 5 6 2 3 23 21 22 24 19 20 17 15 16 18 13 14 11 9 10 12 7 8
12. New Results Theorem 3 (Preston & Whittemore et al.) The following infinite families of graphs (non-unicyclic) have prime vertex labelings: Sn × P4 Sn × P6 Cn × P2 if n + 1 is a prime number
13. New Results Here is an example of the prime labeling for Sn × P6 , specifically 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
14. New Results Theorem 4 (Preston & Whittemore et al.) The family Cn × Pi if n is odd for all i > 1 does not have a prime vertex labeling. 1 5 4 3 2 7 6 10 9 8
15. Future Work Conjectures Here are some infinite families we hope to show have prime vertex labelings: Cn P2 Sk for all n ≥ 3, k ≥ 3 Sn × P12 All unicyclic graphs (Seoud’s Conjecture)
16. Acknowledgements Jeff Rushall and Dana Ernst, advisors Department of Mathematics & Statistics at Northern Arizona University The Center for Undergraduate Research in Mathematics Nebraska Conference for Undergraduate Women in Mathematics Organizers
17. QUESTIONS?
18. Proof An odd-cycle double-tailed tadpole with n vertices is labeled with the following labeling function: f(vi) =          i + 1, for i ≤ k i + 2, for k < i ≤ n − 2 k + 1, for i = n 1, for i = n − 1 Note that v1 must be the vertex adjacent to an intersection point on the even arc length of the cycle and vk is the other intersection point.
19. 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
20. 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.
21. 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
22. Example of Theorem 1 Labeling 1 3 4 2 5 11 13 14 12 15 6 7 9 8 10
23. 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 .
24. Example C4 P2 S5
25. 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
26. General Labeling 6i − 5 6i − 1 6i − 2 6i − 3 6i − 4 6i 6i − 11 6i + 1
27. 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
28. 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.
29. 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
30. 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 pi j , 1 ≤ j ≤ 3. The labeling function f : V → {1, 2, . . . 4n} is given by: f(ci) = 1 i = 1 4i − 1 i ≥ 2 f(pi j ) =          j + 1 i = 1, 1 ≤ j ≤ 3 4i − 3 i ≥ 2, j = 1 4i − 2 i ≥ 2, j = 2 4i i ≥ 2, j = 3
31. Example of a Cycle with Three Hairs 1 2 3 4 15 13 14 16 7 5 6 8 11 9 10 12
32. 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 pi j , 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(pi j ) =      j + 1, 1 ≤ j ≤ 5, i = 1 6(i − 1) + j, 1 ≤ j ≤ 4, i ≥ 2 6(i − 1) + 6, j = 5, i ≥ 2
33. 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
34. 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)}
35. 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
36. 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
37. 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
38. 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
39. Example of C6 × P2 (Hastings Helms) 1 2 3 4 5 6 8 9 10 11 12 7
40. 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.
41. 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 :
42. 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
43. Results Theorem All Sn × P6 are prime.
44. 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
45. 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
46. 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
47. Future Work Hairier Hairy Cycles? Larger Generalized Books? Larger Complete Ternary Trees? Lollipop Graphs? Unrooted Complete Binary Trees? ALL Unicyclic Graphs?????????
48. Acknowledgements Jeff Rushall and Dana Ernst, advisors Department of Mathematics & Statistics at Northern Arizona University The Center for Undergraduate Research in Mathematics
49. QUESTIONS?