Introduction to the Design and Analysis of Algorithms ElSayed Badr Benha University September 20, 2016

Definitions Distance Diameter Radio Labeling Span Radio Number Gear Graph

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

Diameter Diameter: d(G) is the longest distance in a graph G u v

Radio Labeling A Radio Labeling is a one-to-one mapping c: V(G) N satisfying the condition for any distinct vertices (u,v). C: V(G)- N means when you put in a vertices you get out a number. (The vertices are the input to the mapping c, the output consists of natural numbers.) 2 + ≥ 1+3 1 4 8 12 9≥4 u v

Span of a labeling c Span of a labeling c: the max integer that c maps to a vertex of graph G. 1 4 8 12

Radio Number The Radio Number is the lowest span among all radio labelings of a given graph G. Notation: rn(G) = min {rn(c)} 3 4 1 6 1 4 8 12

Radio Number for Path (chain )Graph The labeling from 1 to radio number

Radio Number for Cyclic Graph The labeling from 1 to radio number

Special Graphs for the project 1- Binomial tree 2- Fibonacci tree 3- 4- 5- 6- 7- 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. G7

Theorem: , when n ≥ 7.

Standard labeling for , n odd W2 V3 V2 W3 W1 V4 Z V1 W4 Positions of the the W1 and V1 change W7 V5 V7 W5 W6 V6

Standard labeling for , n even W2 V2 V1 W3 W1 V3 Z V7 W4 W7 V4 V6 W5 W6 V5

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

Lower Bound d(u,v)+ | c(u)-c(v) | ≥ 5 Z 2 3 V W 4 1 Vertex type Strategy: consider placing labels in a manner that omits the fewest values possible. (vertex distance) (label diff) Vertex type Max dist Min diff Z 2 3 V W 4 1 Z W V the chart adds up to 5 exactly what we need to meet our condition

Lower Bound Z 3 2* 1 W n V 2 1** V’s 2(n-1) n-1 Total 2n + 1 4n + 2 Vertices Min label diff Min. # of values omitted Values used Z 3 2* 1 W n V 2 1** V’s 2(n-1) n-1 *Best case: use an extreme value (1 or the span) for Z, otherwise more than two values must be omitted. one **Use the remaining extreme value for one of the V vertices, otherwise more than 1 value must be omitted. other Explain the way the 1 or the span omits less than when the it is labeled in the center. Total 2n + 1 4n + 2 when n ≥ 7.

The Order Of The Pattern W2 X5 X10 V3 V2 X9 W3 X2 W1 X1 X0 Z X11 V4 X8 V1 W4 X6 W7 X4 X12 V5 X14 V7 X7 W6 W5 X3 V6 X13

V W Re-labeling For any given let n =2k or n = 2k+ 1 W2i-1 Xi, Va Xn+a W2i-1 Xi, i= 1,…,k Examples: G7 V5 X7+5 =X12 W2i Xn+k+i i= 1,2,…,k W5 = W2(3) -1 X3 Where i= (a+1)/2 W6 = W2(3) X7+3+3 = X13

{ 3+i 1 ≤ i ≤ n; 2+n+3(i-n) n+1 ≤ i ≤ 2n. X5 8 18 X10 X9 15 Example: 4 X1 X11 2+n+3(i-n) 2+(7)+3( 11 – 7 )= 21 X0 1 X11 21 12 X8 Let’s put the labels on in order from smallest to largest and refer to the formula, e.g. when I is between i and n, each label is just 1 greater than the previous. X6 9 X4 7 24 X12 30 X14 X3 6 X7 10 X13 27

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 18 V2 15 21 1 V1 Case1: u = C (center), v = {V1, …,Vn} * Know c(u) = 1 the possible labels for c(v) = { n+5, n+8,…, 4n+2} Then, d(u,v) = 1 12 Z V1 V2 24 30 27 so, d(u,v) + | c(u) – c(v)| ≥ 1 + |1 - (n +5)| Example: u = Center v = V1 c(u) = 1 c(v) = 12 We will be using v1 since it is the closest radio labeling number to the center. = 1 + n + 4 1 + | 1 - 12 | = n +5 ≥ 5  = 1 + 11 = 12 ≥ 5

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

Conclusion Lower Bound Upper Bound *When n ≥ 7

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).

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

G1 1 4 2

G2 1 2 5 6 3

G3 1 3 6 9 4 7 10

G4 1 8 4 5 14 10 20 17 12

G5 1 4 24 7 18 10 21 15 5 8 12

G6 1 8 12 26 6 23 11 20 5 17 10 14 4