1. 大 家 好 !

2. 完全图的 Cartesian 积的 L(j,k)- 标定 报告人：吕大梅 导师：宋增民 林文松 专业：运筹学与控制论 方向：图论及其应用 学校 : 东南大学

3. L(j,k)-labelings of Cartesian Products of Complete Graphs Speaker: Damei Lü Supervisor: Zengmin Song Advisor: Wensong Lin Speciality: Operational research and cybernetics Field: Graph theory and its application school:southeast university

4. Contents Definitions and Background Cartesian products L(j,k)-labeling Number λ j,k of K n □K m □ K l (n≥m≥l )

5. Definitions and Background L(j,k)-labelings and L(j,k)-labeling number λ j,k ≥ j 0,1, …, t Channel Assignment Problem ≥k

6. Cartesian products Cartesian products of complete graphs Example: K n □K m V( K n □K m )=V( K n ) ×V( K m ) E( K n □K m )={{(a 1, b 1 ), (a 2, b 2 )} | a 1 =a 2 and (b 1, b 2 ) ∈ E( K m ) or b 1 =b 2 and (a 1, a 2 ) ∈ E( K n ) } Preliminary results on Cartesian products of complete graphs

9. L(j,k)-labeling Number λ j,k of K n □ K m □ K l (n≥m≥l ) n=m n ＞ 2mn ＞ 2m n ＝ 2mn ＝ 2m m ＜ n ＜ 2m 3m+2 ＜ 2n ＜ 4m3m+2 ＜ 2n ＜ 4m 2m ＜ 2n≤3m+22m ＜ 2n≤3m+2

10. n ＞ 2m If n ＞ 2m ＞ 4 and j/k≤m then λ j,k ( K n □K m □ K l )=(nm-1)k If n ＞ 2m ＞ 4 and j/k≥m then λ j,k ( K n □K m □ K l )=(n-1)j + (m- 1)k

11. n ＝ 2m If n=2m ＞ 4 and j/k≤m-1 then λ j,k ( K n □K m □ K l )=(nm-1)k If n=2m ＞ 4 and j/k≥m-1 then λ j,k ( K n □K m □ K l )=(n-1)j+(m-1)k

12. 3m+2 ＜ 2n ＜ 4m If 3m+2 ＜ 2n ＜ 4m and j/k≤d=n-m-1 then λ j,k ( K n □K m □ K l )=(nm-1)k If 3m+2 ＜ 2n ＜ 4m and j/k≥d=n-m-1 then λ j,k ( K n □K m □ K l ) ≤ (n-1)[j+(m- d)k]+(m-1)k

13. 2m ＜ 2n≤3m+2 If 2m ＜ 2n ≤ 3m+2 and j/k≤d=[(m+1)/2] then λ j,k ( K n □K m □ K l )=(nm-1)k Suppose m is odd. If 2m ＜ 2n ≤ 3m+2 and j/k≥d=(m+1)/2 then λ j,k ( K n □K m □ K l ) ≤ (n- 1)[j+(m-d)k]+(m-1)k

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