Download presentation

Presentation is loading. Please wait.

Published byClifford Greggs Modified over 2 years ago

2
Definitions Distance Diameter Radio Labeling Span Radio Number Gear Graph

3
Distance Distance: dist(u,v) is the length of a shortest path between u and v in a graph G. uv

4
Diameter Diameter: d(G) is the longest distance in a graph G uv

5
Radio Labeling A Radio Labeling is a one-to-one mapping c: V(G) N satisfying the condition for any distinct vertices (u,v). 14812 uv 2 +≥ 1+3 9≥4

6
Span of a labeling c Span of a labeling c: the max integer that c maps to a vertex of graph G. 14812

7
Radio Number The Radio Number is the lowest span among all radio labelings of a given graph G. Notation: rn(G) = min {rn(c)} 41 6 3 14812

8
Gear Graph A gear graph is a planar connected graph with 2n+1 vertices and 3n edges. The center vertex is adjacent to n vertices which are of degree- three. Between two degree-three vertices is a degree-two vertex. When n≥5 the diameter is 4. G7G7

9
Theorem:, when n ≥ 7.

10
Standard labeling for, n odd Z V1V1 V2V2 V4V4 V3V3 V6V6 V5V5 V7V7 W1W1 W2W2 W3W3 W7W7 W4W4 W5W5 W6W6

11
Standard labeling for, n even Z V7V7 V1V1 V3V3 V2V2 V5V5 V4V4 V6V6 W1W1 W2W2 W3W3 W7W7 W4W4 W5W5 W6W6

12
Prove 1. Define a labeling c 2. Show c is a radio labeling 3. Show span(c) = 4n + 2

13
Lower Bound Vertex type Max distMin diff Z23 V32 W41 d(u,v)+ | c(u)-c(v) | ≥ 5 Z W V (vertex distance) (label diff) Strategy: consider placing labels in a manner that omits the fewest values possible.

14
Lower Bound VerticesMin label diff Min. # of values omitted Values used Z32*1 W10n V21**1 V’s22(n-1)n-1 one other Total2n + 1 4n + 2 when n ≥ 7. *Best case: use an extreme value (1 or the span) for Z, otherwise more than two values must be omitted. **Use the remaining extreme value for one of the V vertices, otherwise more than 1 value must be omitted.

15
ZX0X0 V1V1 V2V2 V3V3 V4V4 V5V5 V6V6 V7V7 X1X1 X2X2 X3X3 X4X4 X5X5 X6X6 X7X7 W1W1 W2W2 W3W3 W4W4 W5W5 W6W6 W7W7 X8X8 X9X9 X 10 X 11 X 12 X 13 X 14 The Order Of The Pattern

16
For any given let n =2k or n = 2k+ 1 W 2i-1 X i, i= 1,…,k W 2i X n+k+i i= 1,2,…,k V a X n+a Re-labeling Examples: G 7 V 5 X 7+5 =X 12 W 5 = W 2(3) -1 X 3 W 6 = W 2(3) X 7+3+3 = X 13 a a W V

17
1 X 13 X 12 X 11 X 10 X9X9 X8X8 X7X7 X1X1 X6X6 X5X5 X4X4 X3X3 X2X2 X 14 1 i = 0; 3+i 1 ≤ i ≤ n; Example: X 1 3+(1) = 4 4 X 11 2+n+3(i-n) 2+(7)+3( 11 – 7 )= 21 { 2+n+3(i-n) n+1 ≤ i ≤ 2n. 21 5 6 7 8 9 10 12 18 15 27 24 30 X0X0

18
Claim: c is a radio labeling for *Note diam(G) = 4 for all when n ≥ 6 WTS: d(u,v) + | c(u) - c(v)| ≥ 1+ diam(G) = 5 Case1: u = C (center), v = {V 1, …,V n } * Know c(u) = 1 the possible labels for c(v) = { n+5, n+8,…, 4n+2} Then, d(u,v) = 1 so, d(u,v) + | c(u) – c(v)| ≥ 1 + |1 - (n +5)| = 1 + n + 4 = n +5 ≥ 5 V1V1 30 27 24 21 18 15 12 Example: u = C enter v = V 1 c(u) = 1 c(v) = 12 1 + | 1 - 12 | = 1 + 11 = 12 ≥ 5 1 Z V1V1 V2V2 V2V2

19
Upper Bound 1 i = 0 3+i 1≤ i ≤ n 2+n+3(i-n) n+1≤ i ≤ 2n { Our goal is to show: n+1≤ i ≤2n2+n+3( - n) i 2 + n + 3n 4n + 2 when n ≥ 7.

20
Conclusion Upper Bound Lower Bound *When n ≥ 7

21
References [1] Chartrand, Erwin, and Zhang, A graph labeling problem suggest by FM channel restrictions, manuscript, 2001. [2] Liu and Zhu, Multi-level distance labeling for paths and cycles, SIAM J. Disc. Math, 2002(revised 2003).

23
Lower Bound Z12 Wn0 V11 Vn-12(n-1) Last other Vertices Total2n + 1 4n + 2 when n ≥ 7. Values used Values omitted The center has a distance of one to all V vertices and a distance of two with W vertices. Every other W vertex has a distance of four. The V vertices have a distance of two between each other. Z d(u,v)+ | c(u)-c(v) | ≥ 5 W V

24
1 42 G1G1

25
1 2 56 3 G2G2

26
1 3 6 9 4 7 10 G3G3

27
1 8 4 5 14 10 20 17 12 G4G4

28
1 4 24 7 18 10 21 15 5 8 12 G5G5

29
1 8 26 6 23 11 20 5 17 10 14 4 G6G6

Similar presentations

OK

Lecture 5 Graph Theory. Graphs Graphs are the most useful model with computer science such as logical design, formal languages, communication network,

Lecture 5 Graph Theory. Graphs Graphs are the most useful model with computer science such as logical design, formal languages, communication network,

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on applications of trigonometry for class 10 download Ppt on minerals and energy resources for class 10 free download Best ppt on history of cricket Ppt on asymptotic notation of algorithms Ppt on history of badminton in the olympics Ppt on recycling of waste product Ppt on culture of punjab Ppt on cross-sectional study define Ppt on job rotation job Ppt online gambling addiction