1 / 24

Radio Labeling of Ladder Graphs

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. Distance: d ( u,v ) Length of shortest path between two vertices u and v Ex:

farica
Download Presentation

Radio Labeling of Ladder Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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.

  2. Distance: d(u,v) Length of shortest path between two vertices u and v Ex: d(v1,v6)=2 Diameter: diam(G) Maximum distance in a graph over all vertices. Ex: diam(G)=3 Graph Terminology V2 V1 V3 V6 V4 V5 G

  3. Radio Labeling • A function c thatassigns positive integer values to each vertex so as to satisfy the radio condition 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

  4. Sample Labeling diam(G)=3 d(u,v) + |c(u) – c(v)| ≥diam(G) + 1 d(u,v)+ |c(u) – c(v)| ≥4 4 1 15 1 1 + |4 – c(v)| ≥ 4 1 + c(v) – 4≥4 c(v)≥7 1 + |1 – c(v)| ≥ 4 1+ c(v) – 1≥4 c(v)≥4 8 4 10 13 2 7 6 10 G Span(c)=10 Yes we can! Span(c) – Maximum label value assigned to a vertex in a graph. Can we get a lower span?

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

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

  7. Lower Bound

  8. Lower Bound Proof: List the vertices of Ln as {x1, x2, …, x2n} in increasing label order: The radio condition implies Rewrite this as

  9. Expansion of the Inequality

  10. Key Idea • c(x2n) is the span of the labeling c. • The smallest possible value of c(x2n) corresponds to the largest possible value of

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

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

  13. Maximizing the Distance

  14. Maximizing the Distance

  15. Maximizing the Distance

  16. Positive Negative Maximizing Distance of L7 Using the best case

  17. Positive Negative Maximizing Distance of L2k+1

  18. Lower Bound for L2k+1

  19. Upper Bound

  20. Labeling Algorithm x3 x6 x13 x12 x9 x10 x1 x11 x5 x8 x4 x7 x2 x15 x21 x17 x25 x19 x23 x14 x24 x26 x16 x20 x22 x18

  21. The Upper Bound • Radio condition: • The upper bound:

  22. Conclusion

  23. Even Ladders

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

More Related