Unicyclic Graphs with Prime Vertex Labelings, II Nathan Diefenderfer and Briahna Preston Joint work with: Michael Hastings, Levi Heath, Hannah Prawzinsky, Emily White & Alyssa Whittemore CURM Whenever we present

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.

Graph Theory Terminology Definition Two vertices are considered adjacent if there is an edge between them. Definition The degree of a vertex is the number of edges having that vertex as an endpoint. Definition A subgraph H of a graph G is a graph whose vertex set is a subset of that of G, and whose adjacency relation is a subset of that of G restricted to this subset.

Infinite Families of Graphs P8 C12 S5

Prime Vertex Labelings 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.

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

Unicyclic Graphs Definition A graph is unicyclic if it contains only one cycle as a subgraph. Conjecture (Seoud and Youssef, 1999) All unicyclic graphs have a prime labeling. 5 1 2 3 4 9 8 10 7 6

Tadpoles Definition A tadpole is a unicyclic graph composed of a cycle and a single path starting at what we call an intersection point, a vertex in the cycle. Definition An arc length is the number of cycle edges between two intersection points. So an even or odd arc length refers to the parity of the number of edges in the arc length.

Double-Tailed Tadpoles Theorem All double-tailed tadpoles have a prime labeling. Labeling - Case 1 The most simple case, a double-tailed tadpole with n vertices and one intersection point vk , is labeled with the following labeling function. f(vi) =      i + 1, for i < k 1, for i = k i, for i > k

A Labeled 6-Cycle Double-Tailed Tadpole 3 4 5 6 1 2 13 12 11 7 8 9 10

Even-Cycle Double-Tailed Tadpole Labeling - Case 2 An even-cycle double-tailed tadpole with n vertices and two distinct intersection points is labeled with the following labeling function: f(vi) = i + 1, for i < n 1, for i = n Note that v1 must be the vertex adjacent to an intersection point.

A Labeled 8-Cycle Double-Tailed Tadpole 4 5 6 7 8 9 2 3 1 15 14 13 10 11 12

Odd-Cycle Double-Tailed Tadpole Labeling - Case 3 An odd-cycle double-tailed tadpole with n vertices and two distinct intersection points 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.

A Labeled 7-Cycle Double-Tailed Tadpole 5 6 7 8 9 2 3 4 1 14 10 11 12 13

Hairy Cycles 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. Definition An m-hairy cycle, denoted Cn Sm is a cycle with m pendants adjacent to each cycle vertex.

Basic Approach For an m-hairy cycle: N is split into sets of m + 1 consecutive integers. Each set is assigned to a cycle vertex and its pendants. A number relatively prime to the rest is assigned to the cycle vertex. Odd m values were used so the number of evens and odds in each set was consistent.

3-Hairy Cycle Theorem For all n ≥ 3, Cn S3 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

A Labeled 3-Hairy Cycle 1 2 3 4 15 13 14 16 7 5 6 8 11 9 10 12

5-Hairy Cycle Theorem For all n ≥ 3, Cn S5 has a prime labeling. 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

A Labeled 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

7-Hairy Cycle Theorem For all n ≥ 3, Cn S7 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 ≤ 7. The labeling formula f : V → {1, 2, . . . 8n} is given by: f(c1) = 1 f(pj 1 ) = j + 1 f(ci) =      8i − 5 i ≡ 2, 3, 6, 8, 9, 11, 12, 14 (mod 15) 8i − 3 i ≡ 4, 5, 7, 10, 13 (mod 15) 8i − 1 i ≡ 0, 1 (mod 15) f(pj i ) ∈ {8i − 7, 8i − 6, . . . , 8i} \ {f(ci)}

A Labeled 7-Hairy Cycle 1 2 3 4 5 6 7 8 19 17 18 20 21 22 23 24 11 9 10 12 13 14 15 16 29 25 26 27 28 30 31 32

Future Work This technique works well for odd m values, so extending this method to evens is an obvious goal. However, this method will fail for m ≥ 16, due to a result by S.S. Pillai, who found that in a string of 17 or more consecutive integers, there will not always be an integer in the set relatively prime to the rest.

Bertrand’s Postulate Prior to this, labelings were constructed for graphs. However, for this graph, a labeling scheme was devised first. Bertrand’s Postulate For every n > 1, there exists a prime p such that n < p < 2n. Unlike hairy cycles, this constructed graph will not have a consistent number of pendants per cycle vertex. Instead, the number will grow to allow us to use Bertrand’s postulate to find primes.

Bertrand Weed Graph Earlier, the natural numbers were split up into sets of a consistent size. For this graph, each set of integers needs to be twice as large as the previous set. Due to Bertrand’s postulate, there is guaranteed to be a prime in each set. Assign the largest prime in each set as the label to the cycle vertex, then label the pendant vertices with the rest of the consecutive integers.

A Labeled Bertrand Weed Graph 1 2 13 10 9 11 14 7 12 8 5 4 3 6

Acknowledgments Center for Undergraduate Research in Mathematics Northern Arizona University Office of Undergraduate Research, NAU Research Advisors Dana Ernst and Jeff Rushall