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Definitions
• Distance
• Diameter
• Radio Labeling
• Span
• Radio Number
• Gear Graph
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DistanceDistance: dist(u,v) is the length of a shortest path
between u and v in a graph G.
u v
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Diameter
Diameter: d(G) is the longest distance in a graph G
u v
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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
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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
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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
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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
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Theorem: , when n ≥ 7.24)( nGrn n
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Standard labeling for , n odd
Z
V1
V2
V4
V3
V6
V5
V7
W1
W2
W3
W7
W4
W5
W6
nG
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Standard labeling for , n even
Z
V7
V1
V3
V2
V5
V4
V6
W1
W2
W3
W7
W4
W5
W6
nG
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Prove 24)( nGrn n
1. Define a labeling c
2. Show c is a radio labeling
3. Show span(c) = 4n + 2
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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.
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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.
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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
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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
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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
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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
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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
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Conclusion
Upper Bound
Lower Bound
24)( nGrn n
24)( nGrn n
24)( nGrn n *When n ≥ 7
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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).
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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
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14 2
G1
4)( 1 Grn
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1
2
56
3
G2
6)( 2 Grn
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1
3
6
9
4
7
10
G3
10)( 3 Grn
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1
8
45
14
10
20
17
12
G4
20)( 4 Grn
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1
4
24
7
18
10
21
15
5
8
12G5
23)( 5 Grn
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1
8
12
26
623
11
20
5
17
1014
4
G6
26)( 6 Grn