Radio Labeling Cartesian Products of Path Graphs

1. Radio Labeling Cartesian Products of Path Graphs Eduardo Calles and Henry Gómez Advisors: Drs. Maggy Tomova and Cindy Wyels Funding: NSF, NSA, and Moody’s, via the SUMMA program.

2. Grid Graphs • The path graph Pn contains n consecutive vertices connected along a sequence of n - 1 edges. • When we “multiply” two path graphs (take the Cartesian product), the graph product is a grid graph. Ex: P5 Ex: P3 □ P3

3. Transition to Tables v3 v2 v1 v1 v2 v3 v4 v6 v5 v4 v5 v6 v7 v8 v9 v9 v8 v7 Vertices are now represented by boxes. Two boxes represent adjacent vertices if they share an edge.

4. Strategy for Establishing an Upper Bound for rn(Pn□Pn) • Specify the order in which we’ll label vertices. • Give the vertices the minimum label values required so as to satisfy the radio condition. • Calculate the span of this labeling. • This span is an upper bound for rn(Pn□Pn).

5. 4 16 36 64 73 50 26 10 2 14 34 62 75 11 71 3 52 28 12 10 2 32 60 77 13 69 54 30 12 58 79 67 56 81 65 72 51 63 74 42 70 53 61 76 44 20 40 68 55 59 78 46 22 6 18 38 66 57 80 48 24 8 2 3 Order of Vertex Labels 4 16 36 43 26 10 2 14 4 34 45 16 21 41 10 28 2 12 32 47 14 23 4 7 19 2 12 39 30 49 25 9 1 5 17 37 42 27 20 11 6 3 15 8 35 22 44 20 40 6 29 18 13 24 33 46 8 22 6 18 38 31 48 24 8

6. General Ordering (for Odds) 1,1 1,k 1,k+1 1,2k+1 Stage 0: (k+1, k+1) Stage 1: top right side, top left side, bottom left side, bottom right side, Stage 2: repeat the cycle k = number of stages Any stages greater than stage 0 complete the diamond cycle pattern k,1 k+1,1 k+1,k+1 2k+1,1 2k+1,2k+1 2k+1,k 2k+1,k+1 Distance traveled to start new stage is 2k; Distance traveled within stages is 2k+1 each time.

7. Calculating the Span for Odd Graphs

8. Strategy to Establish a Lower Bound for rn(Pn□Pn) Let c be any radio labeling of Pn □ Pn. • Develop equation relating span(c) to the sum of distances between consecutively-labeled vertices. • Minimize the span by maximizing this sum of distances. This minimum span is a lower bound for rn(Pn□Pn).

9. General Equation for span(c) List the vertices of Pn□ Pnas {x1,…,xn2} in increasing label order: • Radio Condition gives a necessary condition: • Rewrite:

10. Expansion of the Inequality To minimize the span, maximize the sum of the distances.

11. x2 = v(1,2) x3 = v(1,3) x1 = v(1,1) x4 =v(2,1) x6 = v(2,3) x5 = v(2,2) x9 = v(3,3) x7 = v(3,1) x8 = v(3,2) Calculating Distance d(xi,xi+1) What is the distance between x3 and x4?

12. Maximizing the Sum of Distances

13. Maximizing the Sum of Distances By examining the sum of distances, σ’s and τ’s appear

14. Maximizing the Sum of Distances n = 2k+1 Positive Negative Same amount need to be added and subtracted.

15. Generalized Lower Bound for Odds where n = 2k + 1

16. Our Results • Odds ( n = 2k + 1) • Evens ( n = 2k)