Prime Vertex Labelings of Graphs

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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

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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.

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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.

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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 .

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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

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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, ...

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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, ...

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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.

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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

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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

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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

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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

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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

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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

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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)

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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

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QUESTIONS?

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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.

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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

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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.

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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

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

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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 .

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Example C4 P2 S5

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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

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

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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

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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.

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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

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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

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

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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

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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

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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)}

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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

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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

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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

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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

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

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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.

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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 :

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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

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Results Theorem All Sn × P6 are prime.

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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

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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

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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

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

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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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QUESTIONS?