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# Unicyclic Graphs with Prime Vertex Labelings, I

This talk was given by my undergraduate research students Levi Heath and Emily White on March 20, 2015 at the 2015 MAA/CURM Spring Conference at Brigham Young University.

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

March 20, 2015

## Transcript

1. ### Unicyclic Graphs with Prime Vertex Labelings, I Levi Heath and

Emily White Joint work with: Nathan Diefenderfer, Michael Hastings, Hannah Prawzinsky, Briahna Preston & Alyssa Whittemore 2015 MAA/CURM Spring Conference March 20, 2015
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.
3. ### 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.
4. ### Connected and Unicyclic 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. Deﬁnition A unicyclic graph is a simple graph containing exactly one cycle. Here is a unicyclic graph that is NOT connected.
5. ### Pendants Deﬁnition In a unicyclic graph, a pendant is a

path on two vertices with exactly one vertex being a cycle vertex. The non-cycle vertex of a pendant is called a pendant vertex.

7. ### 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 .
8. ### 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
9. ### Prime Vertex Labeling Deﬁnition An n-vertex graph has a prime

vertex labeling if its vertices are labeled with the integers 1, 2, 3, . . . , n such that no label is repeated and all 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 Some useful number theory facts: All pairs of consecutive integers are relatively prime. Consecutive odd integers are relatively prime. A common divisor of two integers is also a divisor of their difference. The integer 1 is relatively prime to all integers.
10. ### Conjecture Conjecture (Seoud and Youssef, 1999) All unicyclic graphs have

a prime vertex labeling.
11. ### Known Prime Vertex Labelings 1 2 3 4 5 6

7 8 P8 1 12 11 10 9 8 7 6 5 4 3 2 C12 1 2 6 5 4 3 S5
12. ### New Inﬁnite Families of Graphs Deﬁnition A cycle pendant star

is a cycle with each cycle vertex adjacent to an identical star, denoted Cn P2 Sk (for n ≥ 3). Deﬁnition A generalized cycle pendant star is a cycle with each cycle vertex adjacent to an identical star and each noncentral star vertex is a central vertex for an identical star, denoted Cn P2 Sk Sk Sk · · · Sk (for n ≥ 3).

14. ### Prime Vertex Labelings of Inﬁnite Families of Graphs Theorem For

all n ≥ 3, we have: The cycle pendant stars, Cn P2 Sk are prime for 3 ≤ k ≤ 8. The generalized cycle pendant star, Cn P2 S3 S3 has a prime vertex labeling. The cases for k ≥ 9 for cycle pendant stars and all other cases of generalized cycle pendant stars remain open.
15. ### Labeling Function of Cn P2 S3 Labeling Function Let c1,

c2, . . . , cn denote the cycle labels, p1, p2, . . . , pn denote the vertices adjacent to the corresponding cycle vertices and the pendant vertices adjacent to 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
16. ### Prime Vertex Labeling of C3 P2 S3 1 3 4

2 5 11 13 14 12 15 6 7 9 8 10
17. ### Generalized Labeling of Cn P2 S4 6i − 5 6i

− 1 6i − 4 6i − 3 6i − 2 6i 6i − 11 6i + 1
18. ### Prime Vertex Labeling of C4 P2 S4 1 5 2

3 4 6 19 23 20 21 22 24 7 11 8 9 10 12 13 17 14 15 16 18
19. ### 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

21. ### Prime Vertex Labeling of C4 P2 S7 1 5 2

3 4 6 7 8 9 10 13 11 12 14 15 16 17 18 19 23 20 21 22 24 25 26 27 28 31 29 30 32 33 34 35 36
22. ### Prime Vertex Labeling of C4 P2 S3 S3 1 2

5 9 11 3 4 6 7 8 10 12 13 14 15 16 19 23 25 17 18 20 21 22 24 26 27 28 29 32 31 35 41 30 33 34 36 37 38 39 40 42 43 44 47 51 53 45 46 48 49 50 52 54 55 56
23. ### Future Work Conjecture For all n, k ∈ N with

n ≥ 3, Cn P2 Sk is prime. Conjecture For all n, k ∈ N with n ≥ 3, Cn P2 Sk Sk Sk · · · Sk is prime.
24. ### Acknowledgments Center for Undergraduate Research in Mathematics Northern Arizona University

Research Advisors Dana Ernst and Jeff Rushall
25. ### Labeling Function of Cn P2 S3 S3 Note that Cn

P2 S3 S3 contains 14n vertices. We will identify our vertices as follows. Let ci, 1 ≤ i ≤ n denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , let the non-cycle vertices adjacent to pi be denoted si,j for 1 ≤ j ≤ 3, and let the remaining vertices adjacent to each si,j be denoted li,j,k for 1 ≤ k ≤ 3. Our labeling function f : V → {1, 2, . . . , 14n} is best deﬁned by ﬁrst describing cycle and pendent vertex labels: f(ci) = 14i − 13, 1 ≤ i ≤ n f(pi) = 14i − 12, i ≡3 1, 2 14i − 10, i ≡3 0
26. ### Labeling Function of Cn P2 S3 S3 The remaining vertex

labels are determined by the values of i, j and k as follows. If i ≡3 1, 2, then deﬁne f(si,j) =      14i − 9, j = 1 14i − 5, j = 2 14i − 3, j = 3 f(li,j,k ) =                                  14i − 11, j = 1, k = 1 14i − 10, j = 1, k = 2 14i − 8, j = 1, k = 3 14i − 7, j = 2, k = 1 14i − 6, j = 2, k = 2 14i − 4, j = 2, k = 3 14i − 2, j = 3, k = 1 14i − 1, j = 3, k = 2 14i, j = 3, k = 3
27. ### Labeling Function of Cn P2 S3 S3 If i ≡3

0, then deﬁne f(si,j) =      14i − 11, j = 1 14i − 7, j = 2 14i − 1, j = 3 f(li,j,k ) =                                  14i − 12, j = 1, k = 1 14i − 9, j = 1, k = 2 14i − 8, j = 1, k = 3 14i − 6, j = 2, k = 1 14i − 5, j = 2, k = 2 14i − 4, j = 2, k = 3 14i − 3, j = 3, k = 1 14i − 2, j = 3, k = 2 14i, j = 3, k = 3
28. ### Labeling Function of Cn P2 S4 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
29. ### Labeling Function of Cn P2 S5 Let ci , 1

≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 5, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 7n} is given by: f(ci) = 7i − 6, 1 ≤ i ≤ n f(pi) =                7i − 2, i ≡6 1, 3 7i − 3, i ≡6 2, 4 7i − 4, i ≡6 5 7i − 5, i ≡6 0, i ≡30 0 7i − 1, i ≡30 0
30. ### Labeling Function of Cn P2 S5 f(oi1) = 7i −

5, i ≡6 0 or i ≡30 0 7i − 4, i ≡6 0, i ≡30 0 f(oi2) = 7i − 4, i ≡6 0, 5 or i ≡30 0 7i − 3, i ≡6 5 or i ≡6 0, i ≡30 0 f(oi3) = 7i − 3, i ≡6 0, 2, 4, 5 or i ≡30 0 7i − 2, i ≡6 1, 3 f(oi4) = 7i − 2, i ≡30 0 7i − 1, i ≡30 0 f(oi5) = 7i, 1 ≤ i ≤ n
31. ### Labeling Function of Cn P2 S6 Let ci , 1

≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 6, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 8n} is given by: f(ci) = 8i − 7, 1 ≤ i ≤ n f(pi) =      8i − 3, i ≡3 0 8i − 5, i ≡3 0, i ≡15 0 8i − 1, i ≡15 0
32. ### Labeling Function of Cn P2 S6 f(oi1) = 8i −

6, 1 ≤ i ≤ n f(oi2) = 8i − 5, i ≡3 0 or i ≡15 0 8i − 3, i ≡3 0, i ≡15 0 f(oi3) = 8i − 4, 1 ≤ i ≤ n f(oi4) = 8i − 3, i ≡15 0 8i − 1, i ≡15 0 f(oi5) = 8i − 2, 1 ≤ i ≤ n f(oi6) = 8i, 1 ≤ i ≤ n
33. ### Labeling Function of Cn P2 S7 Let ci , 1

≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 7, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 9n} is given by: f(ci) = 9i − 8, 1 ≤ i ≤ n f(pi) =          9i − 4, i ≡2 1 9i − 5, i ≡2 0, i ≡10 0 9i − 7i ≡10 0, i ≡70 0 9i − 1, i ≡70 0
34. ### Labeling Function of Cn P2 S7 f(oi1) = 9i −

7, i ≡10 0 or i ≡70 0 9i − 5, i ≡10 0, i ≡70 0 f(oi2) = 9i − 6, 1 ≤ i ≤ n f(oi3) = 9i − 5, i ≡2 1 or i ≡70 0 9i − 4, i ≡2 0, i ≡70 0 f(oi4) = 9i − 4, i ≡70 0 9i − 1, i ≡70 0 f(oi5) = 9i − 3, 1 ≤ i ≤ n f(oi6) = 9i − 2, 1 ≤ i ≤ n f(oi7) = 9i, 1 ≤ i ≤ n
35. ### Labeling Function of Cn P2 S8 Let ci , 1

≤ i ≤ n, denote the cycle vertices, let pi denote the pendant vertex adjacent to ci , and let oik , 1 ≤ k ≤ 8, denote the outer vertices adjacent to pi . The labeling function f : V → {1, 2, . . . 10n} is given by: f(ci) = 10i − 9, 1 ≤ i ≤ n f(pi) =      10i − 3, i ≡3 0 10i − 7, i ≡3 0, i ≡21 0 10i − 1, i ≡21 0
36. ### Labeling Function of Cn P2 S8 f(oi1) = 10i −

8, 1 ≤ i ≤ n f(oi2) = 10i − 7, i ≡21 0 or i ≡3 1, 2 10i − 3, i ≡3 0, i ≡21 0 f(oi3) = 10i − 6, 1 ≤ i ≤ n f(oi4) = 10i − 5, 1 ≤ i ≤ n f(oi5) = 10i − 4, 1 ≤ i ≤ n f(oi6) = 10i − 3, i ≡21 0 10i − 1, i ≡21 0 f(oi7) = 10i − 2, 1 ≤ i ≤ n f(oi8) = 10i, 1 ≤ i ≤ n