Labeling graphs with two distance constraints

Chang, Hsun-Wen; Chou, Huang-Wei; Kuo, Dong‐Hau; Lin, Chih‐Hong

doi:10.1016/j.disc.2007.10.036

Cited by 1 publication

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“…Proof: By Theorem 13 of Lin (2002) also in Chang et al (2006) and Corollary 1. We first consider the case 3 ≤ n ≤ 2d 1 + 1.…”

Section: The Lower Bounds Of Assignment Numbersmentioning

confidence: 95%

“…In 2002, Lin further generalised to study the L(d 1 , d 2 ; l 1 , l 2 )-assignment problem (Lin, 2002) of paths and Cartesian of two paths. During several decades, the problem has attracted many researchers (Some of them are: Bertossi et al, 2003;Chang and Kuo, 1996;Chang and Lu, 2003;Chang et al, 2000Chang et al, , 2006Chou, 2003;Georges and Mauro, 2002;Georges et al, 1994;Heuvel et al, 1998;McDiarmid, 2003;van den Havet, 2001;West, 2001;Whittlesey et al, 1995;Yeh, 2006).…”

Section: Introductionmentioning

confidence: 99%

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Optimal frequency assignments of cycles and powers of cycles

Chang

Chen

2006

IJMNDI

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To reduce the interfering among transmitters, any two 'close' transmitters, with distance no more than d 2 , must receive frequencies at least l 2 apart, and any two 'very close' transmitters, with distance no more than d 1 , must receive channels at least l 1 apart. The assignment problem is to assign a frequency to each transmitter such that total bandwidth of assigned frequencies can be minimised. In this paper, we present upper and lower bounds of the minimum bandwidth of cycles and powers of cycles. We also show the corresponding assignment functions. And then we partially solve the assignment problem of cycles C n and their powers such as: (i) 3 ≤ n ≤ 2d 1 + 1; (ii) 2d 1 + 2 ≤ n ≤ 2d 2 + 1, q 1 = 0 and l 1 ≤ m 1 l 2 ; (iii) 2d 1 + 2 ≤ n ≤ 2d 2 + 1, q 1 ≡ 1(mod m 1 ) and l 1 > m 1 l 2 ; (iv) 2d 1 + 2 ≤ n ≤ 2d 2 + 1, q 1 = 0 and l 1 ≥ (m 1 + d 1 − 1)l 2 or l 1 ≤ (m 1 − 1)l 2 ;(v) n ≥ 2d 2 + 2 and l 1 ≤ (d 2 + q 2 /m 2 )/(d 1 + 1) l 2 ; and (vi) n ≥ 2d 2 + 2, q 1 ≡ 1(mod m 1 ), and l 1 > m 1 l 2 , where n = m i (d i + 1) + q i and 0 ≤ q i ≤ d i .

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“…Proof: By Theorem 13 of Lin (2002) also in Chang et al (2006) and Corollary 1. We first consider the case 3 ≤ n ≤ 2d 1 + 1.…”

Section: The Lower Bounds Of Assignment Numbersmentioning

confidence: 95%