Download presentation

Presentation is loading. Please wait.

Published bySteven Gilstrap Modified over 2 years ago

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(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

3
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

4
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

5
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)

6
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

7
Lower Bound

8
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

9
Expansion of the Inequality

10
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

11
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)

12
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)

13
Maximizing the Distance

16
Positive Negative Maximizing Distance of L Maximizing Distance of L 7 Using the best case

17
Maximizing Distance of L Maximizing Distance of L 2k+1 Positive Negative

18
Lower Bound for L 2k+1

19
Upper Bound

20
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

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.

Similar presentations

OK

Can Dijkstra’s Algorithm be modified in an obvious way to give the longest path in a graph? a). Yes b). No c). I have absolutely no idea.

Can Dijkstra’s Algorithm be modified in an obvious way to give the longest path in a graph? a). Yes b). No c). I have absolutely no idea.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on idiopathic thrombocytopenia purpura signs Ppt on question tags rules Ppt on teachers day message Ppt on time management techniques Ppt on manufacturing industry in india Ppt on conservation of electricity Ppt on suspension type insulation tool Ppt on beer lambert law spectrophotometry Ppt on grease lubrication lines Ppt on gender inequality in education