Definitions

• Distance

• Diameter

• Radio Labeling

• Span

• Radio Number

• Gear Graph

DistanceDistance: 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 )(1)()(),( Gdiamvcucvud

for any distinct vertices (u,v).

1 4 8 12

u v

)(1)()(),( Gdiamvcucvud

2 + 81 ≥ 1+3

9≥4

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)}

4 1 6 3

1 4 8 12

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 ≥ 7.24)( nGrn n

Standard labeling for , n odd

Z

V1

V2

V4

V3

V6

V5

V7

W1

W2

W3

W7

W4

W5

W6

nG

Standard labeling for , n even

Z

V7

V1

V3

V2

V5

V4

V6

W1

W2

W3

W7

W4

W5

W6

nG

Prove 24)( nGrn n

1. Define a labeling c

2. Show c is a radio labeling

3. Show span(c) = 4n + 2

Lower Bound

Vertex

type

Max dist Min diff

Z 2 3

V 3 2

W 4 1

d(u,v)+ | c(u)-c(v) | ≥ 5

ZW

V

(vertex distance)

(label diff)

Strategy: consider placing labels in a manner that omits the fewest values possible.

Lower Bound

Vertices Min label diff

Min. # of values omitted

Values used

Z 3 2* 1

W 1 0 n

V 2 1** 1

V’s 2 2(n-1) n-1

one

other

Total 2n + 1 2n + 1

4n + 2

when n ≥ 7.nn GnGrn 24)(

*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.

ZX0

V1

V2V3

V4

V5

V6

V7

X1

X2

X3

X4

X5

X6

X7

W1

W2

W3

W4

W5 W6

W7

X8

X9X10

X11

X12

X13

X14

The Order Of The Pattern

For any given let n =2k or n = 2k+ 1

•W2i-1 Xi,

i= 1,…,k

•W2i Xn+k+i

i= 1,2,…,k

•Va Xn+a

Re-labeling

Examples: G7

V5 X7+5 =X12

W5 = W2(3) -1 X3

W6 = W2(3) X7+3+3 = X13

nG

xaa

a

W

V

1

X13

X12

X11

X10

X9

X8

X7

X1

X6

X5

X4

X3

X2

X14

)( iXc 1 i = 0;

3+i 1 ≤ i ≤ n;

Example:

X1 3+(1) = 4

4

X11 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

1815

27

24 30

X0

7G

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 = {V1, …,Vn}

* 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

V1

30

27

24

21

1815

12

Example: u = Center v = V1

c(u) = 1 c(v) = 12

7G

1 + | 1 - 12 |

= 1 + 11

= 12 ≥ 5

1Z

V1 V2

nG

nGV2

Upper Bound

)( iXc1 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

24)( nGrn n

when n ≥ 7.24)( nGrn n nG

Conclusion

Upper Bound

Lower Bound

24)( nGrn n

24)( nGrn n

24)( nGrn n *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 0

V 1 1

V n-1 2(n-1)

Last

other

Vertices

Total 2n + 1 2n + 1

4n + 2

when n ≥ 7.nn GnGrn 24)(

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

14 2

G1

4)( 1 Grn

1

2

56

3

G2

6)( 2 Grn

1

3

6

9

4

7

10

G3

10)( 3 Grn

1

8

45

14

10

20

17

12

G4

20)( 4 Grn

1

4

24

7

18

10

21

15

5

8

12G5

23)( 5 Grn

1

8

12

26

623

11

20

5

17

1014

4

G6

26)( 6 Grn