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Radio Labeling of Ladder Graphs Josefina Flores Kathleen Lewis From: California State University Channel Islands Advisors: Dr. Tomova and Dr.Wyels Funding: NSF, NSA, and Moody’s, via the SUMMA program.

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Distance: d(u,v) Length of shortest path between two vertices u and v Ex: d(v 1,v 6 )=2 Diameter: diam(G) Maximum distance in a graph over all vertices. Ex: diam(G)=3 Graph Terminology V1V1 V2V2 V3V3 V4V4 V5V5 V6V6 G

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Radio Labeling A function c that assigns positive integer values to each vertex so as to satisfy the radio conditionA function c that assigns positive integer values to each vertex so as to satisfy the radio condition d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. d(u,v) + |c(u)-c(v)| ≥ diam(G) + 1. diam (G) - diameter of graph d(u,v) - distance between vertices u and v c(u),c(v) – label assigned to vertices c(u),c(v) – label assigned to vertices

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1 + |1 – c(v)| ≥ 4 1 + c(v) – 1 ≥ 4 c(v) ≥ 4 c(v) ≥ 4 d(u,v) + |c(u) – c(v)| ≥ 4 1 41 84 27610 13 15 1 + |4 – c(v)| ≥ 4 1 + c(v) – 4 ≥ 4 c(v) ≥ 7 c(v) ≥ 7 10 G Sample Labeling d(u,v)d(u,v)|c(u) – c(v)| 13 22 31 Span – Maximum label value assigned to a vertex in a graph. Span(c) – Maximum label value assigned to a vertex in a graph. diam(G)=3 Can we get a lower span? Span(c)=10 Yes we can! d(u,v) + |c(u) – c(v)| ≥ diam(G) + 1

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4 10 1 2 6 1 13 15 7 10 4 8 What is Radio Number? The radio number of G, rn(G), is the minimum span, taken over all possible radio labelings of G. G rn(G)

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V (2,5) V (1,2) V (1,1) V (1,3) V (1,4) V (1,5) V (1,6) V (1,7) V (2,2) V (2,1) V (2,3) V (2,4) V (2,6) V (2,7) V (1,2) V (2,5) What is the distance between V (1,2) and V (2,5) ? Odd Ladders

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Lower Bound

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Proof: List the vertices of L n as {x 1, x 2, …, x 2n } in increasing label order: The radio condition implies Rewrite this as

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Expansion of the Inequality

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Key Idea c(x 2n ) is the span of the labeling c. The smallest possible value of c(x 2n ) corresponds to the largest possible value of

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σ-τ Notation V (1,1) V (1,2) V (1,3) V (1,4) V (1,5) V (1,6) V (1,7) V (2,2) V (2,1) V (2,3) V (2,4) V (2,5) V (2,6) V (2,7)

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Maximizing the Distance V (1,n) V (1,1) V (1,2) V (1,k) V (1,k+1) V (2,2) V (2,1) V (2,k) V (2,k+1) V (1,n-1) V (2,n-1) V (2,n)

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Maximizing the Distance

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Positive Negative Maximizing Distance of L Maximizing Distance of L 7 Using the best case

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Maximizing Distance of L Maximizing Distance of L 2k+1 Positive Negative

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Lower Bound for L 2k+1

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Upper Bound

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Labeling Algorithm x3x3x3x3 x 12 x6x6x6x6 x8x8x8x8 x 10 x1x1x1x1 x4x4x4x4 x 13 x7x7x7x7 x9x9x9x9 x 11 x2x2x2x2 x5x5x5x5 x 14 x 15 x 16 x 17 x 18 x 19 x 20 x 21 x 22 x 23 x 24 x 25 x 26

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The Upper Bound Radio condition: The upper bound:

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Conclusion

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Even Ladders

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References D. Liu and X. Zhu, Multilevel Distance labelings for paths and cycles, SIAM J. Discrete Math. 19 (2005), No. 3, 610-621.

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