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Definitions Distance Diameter Radio Labeling Span Radio Number Gear Graph

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

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

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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

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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. 14812

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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)} 41 6 3 14812

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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. G7G7

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Theorem:, when n ≥ 7.

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

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

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Prove 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 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.

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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.

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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

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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

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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

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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 = {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

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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.

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Conclusion Upper Bound Lower Bound *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 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

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1 42 G1G1

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1 2 56 3 G2G2

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1 3 6 9 4 7 10 G3G3

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1 8 4 5 14 10 20 17 12 G4G4

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

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

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