Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan.

Transcription

1 Gaphs and Cobinatoics (1999) 15 : 95±301 Gaphs and Cobinatoics ( Spinge-Veag 1999 T-Cooings and T-Edge Spans of Gaphs* Shin-Jie Hu, Su-Tzu Juan, and Gead J Chang Depatent of Appied Matheatics, Nationa Chiao Tung Univesity, Hsinchu 30050, Taiwan e-ai: gjchang@athnctuedutw Abstact Suppose G is a gaph and T is a set of non-negative integes that contains 0 A T-cooing of G is an assignent of a non-negative intege f x to each vetex x of G such that j f x f y j B T wheneve xy A E G The edge span of a T-cooing f is the axiu vaue of j f x f y j ove a edges xy, and the T-edge span of a gaph G is the iniu vaue of the edge span of a T-cooing of G This pape studies the T-edge span of the dth powe Cn d of the n-cyce C n fo T ˆf0; 1; ; ; g In paticua, we nd the exact vaue of the T-edge span of Cn d fo n 1 0 o 1 (od d 1), and owe and uppe bounds fo othe cases 1 Intoduction T-cooings wee intoduced by Hae [3] in connection with the channe assignent pobe in counications In this pobe, thee ae n tansittes x 1 ; x ; ; x n situated in a egion We wish to assign to each tansitte x a fequency f x Soe of the tansittes intefee because of poxiity, eteooogica, o othe easons To avoid intefeence, two intefeing tansittes ust be assigned fequencies such that the absoute di eence of thei fequencies does not beong to the fobidden set T of non-negative integes and T contains 0 The objective is to ae a fequency assignent that is e½cient accoding to cetain citeia, whie satisfying the above constaint To fouate the channe assignent pobe gaph-theoeticay, we constuct a gaph G in which V G ˆfx 1 ; x ; ; x n g, and thee is an edge between tansittes x i and x j if and ony if they intefee Given gaph G and a set T of non-negative integes and T contains 0, a T-cooing of G is a function f fo V G to the set of non-negative integes such that xy A E G ipies j f x f y j B T: Fo the case when T ˆf0g, T-cooing is the odinay vetex cooing In channe assignents, the objective is to aocate the channes e½cienty Fo the T-cooing standpoint, thee citeia ae ipotant fo easuing the * Suppoted in pat by the Nationa Science Counci unde gant NSC85-11-M009-04

2 96 S Hu et a e½ciency: st, the ode of a T-cooing, which is the nube of di eent coos used in f; second, the span of f, which is the axiu of j f x f y j ove a vetices x and y; and thid, the edge span of f, which is the axiu of j f x f y j ove a edges xy Given T and G, the T-choatic nube w T G is the iniu ode of a T-cooing of G, the T-span sp T G is the iniu span of a T-cooing of G, and the T-edge span esp T G is the iniu edge span of a T-cooing of G Cozzens and Robets [1] showed that the T-choatic nube w T G is equa to the choatic nube w G, which is the iniu nube of coos needed to coo the vetices of G so that adjacent vetices have di eent coos The paaete T-span of gaphs has been studied extensivey; fo a good suvey, see [6]; fo ecent esuts, see [, 5, 7] Howeve, copaing to T-spans, thee ae eativey fewe nown esuts about T-edge spans of gaphs, see [1, 4] Cozzens and Robets [1] aised the pobe of coputing T-edge spans of non-pefect gaphs when T ˆf0; 1; ; ; g Liu [4] studied this pobe fo odd cyces In this atice, we conside Cn d,thedth powe of the n-cyce C n The gaph Cn d has the vetex set V C n d ˆfv 0; v 1 ; ; v n 1 g and the edge set E Cn d ˆ 6 fv i v j : j ˆ i 1; i ; ; i dg; 0UiUn 1 whee the index j fo v j is taen oduo n We nd the exact vaue of esp T Cn d fo n 1 0 o 1 (od d 1), and owe and uppe bounds fo othe cases Pevious esuts In this section, we quote soe nown esuts about T-spans and T-edge spans, soe of which wi be used in Section 3 The cique nube o G of G is the axiu ode of a cique (copete gaph), a set of paiwise adjacent vetices A copete gaph of ode n is denoted by K n The n-cyce is the gaph C n with vetex set V C n ˆfv 0 ; v 1 ; ; v n 1 g and edge set E C n ˆfv 0 v 1 ; v 1 v ; ; v n v n 1 ; v n 1 v 0 g Note that C 1 n is C n The foowing ae soe nown esuts on T-spans and T-edge spans Theoe 1 (Cozzens and Robets [1]) The foowing stateents hod fo a gaphs G and sets T (1) w G 1Uesp T G U sp T G () sp T K og U esp T G U sp T G U sp T K w G (3) If T is -initia, ie, T ˆf0; 1; ; g U S whee S contains no utipe of, then sp T G ˆsp T K w G ˆ w G 1 Theoe (Liu [4]) Fo any odd cyce C n and T ˆf0; 1; ; g; n 1 esp T C n ˆ n Figue 1 shows an exape of C n with T ˆf0; 1; g fo which w T C 7 ˆ 3 < esp T C 7 ˆ4 < sp T C 7 ˆ6 These vaues foow fo Theoes 1 and

3 T-Cooings and T-Edge Spans of Gaphs 97 Fig 1 C 7 with T ˆf0; 1; g 3 Edge spans fo powes of n-cyces This section gives esuts fo T-edge spans of Cn d fo the -initia set T ˆf0; 1; ; ; g j We note that Cn d G K n fo d V n and esp T K n ˆsp T K n j ˆ n 1 Theefoe, thoughout this atice we conside Cn d ony fo d U n and assue n ˆ d 1, whee V and 0 U U d Ou ain esuts ae as foows Fist, we give an uppe bound and a owe bound fo esp T Cn d (Theoe 4), both of the ipy the exact vaue of esp Cn d when ˆ 0 (Theoe 5) We then give a bette uppe bound when gcd n; d 1 ˆ1 (Theoe 6) and a bette owe bound when V 1 (Theoe 7), both of the ipy the exact vaue when ˆ 1 (Theoe 8) Lea 3 If n ˆ d 1 with V and 0 U U d, then w Cn d ˆd 1 and w Cn d ˆ n ˆ d 1 j Poof It is easy to see that o Cn d ˆd 1 since d 1 U n ; and w Cn d V n since any independent set of Cn d contains at ost vetices Letting n i ˆ n i, we have n ˆ X 1 n i : Coo the n vetices of Cn d as 1; ; ; n 0; 1; ; ; n 1 ; 1; ; ; n ; ; 1; ; ; n 1 This cooing is a pope vetex cooing since each n i V n 1 ˆ d 1 and so n i V d 1 Hence w Cn d U n Theoe 4 If n ˆ d 1 with V and 0 U U d, then d U esp T Cn d U sp T Cn d ˆd Poof The theoe foows fo Theoe 1 and Lea 3

4 98 S Hu et a Theoe 5 If n ˆ d 1 with V, then esp T Cn d ˆsp T Cn d ˆd Poof The theoe foows fo Theoe 4 as ˆ 0 Theoe 6 Suppose n ˆ d 1 with V and 0 U U d If gcd n; d 1 ˆ1, then esp T Cn d U d Poof Since gcd n; d 1 ˆ1; d 1 is a geneato of Z n using oduo n addition, ie, j i 1 i d 1 (od n) fo 0U i U n geneates each intege in f0; 1; ; n g exacty once In othe wods, we can conside V Cn d as fv j0 ; v j1 ; ; v jn 1 g Note that any cicuay consecutive vetices v ja 1 ; v ja ; ; v ja (with indices a p consideed oduo n) fo an independent set in Cn d Consequenty, v j a v jb is not an edge when 0 U a < b U n with 1 U infb a; n a bg U Now, conside the function f on V Cn d de ned by f v j i ˆ i fo 0 U i U n We cai that f is a T-cooing Fo any edge v ja v jb with 0 U a < b U n, accoding to the peceding discussion, infb a; n a bg V, ie, U b a U n ˆ d Then 8 j f v ja f v jb j ˆ b a V b a >< V 1 ; b U a d >: U 1 ; o 8 V ; >< j f v ja f v jb j U d >: : Theefoe, f is a T-cooing of Cn d and esp T Cn d U d Theoe 7 If n ˆ d 1 with V and 1 U U d, then esp T Cn d V d Poof Suppose esp T Cn d U d Let f be a T-cooing fo which esp T Cn d ˆaxfj f v i f v j j : v i v j A E Cn d g Note that the 1 vetices v i d 1 ; 0 U i U, ae paiwise non-adjacent except fo v 0 v d 1 A E Cn d Let e i; j ˆ f v i d 1 f v j d 1 fo 0 U i U j U Then U je 0; jˆ X 1 e i; i 1 U X 1 je i; i 1 j and so thee exists at east one i such that je i; i 1 j V In othe wods, the set U ˆfi : je i; i 1 j V and 0 U i U g is not epty

5 T-Cooings and T-Edge Spans of Gaphs 99 Fo any i A U, the d vetices v j ; i d 1 U j U i 1 d 1 ; ae paiwise adjacent except that v i d 1 is not adjacent to v i 1 d 1 Sot the d vaues f v j ; i d 1 U j U i 1 d 1 ; into b 1 U b U U b d If fb 1 ; b d g 0 f f v i d 1 ; f v i 1 d 1 g, then esp Cn d V b d b 1 ˆ Xd 1 b j 1 b j V d ; a contadiction Hence, fb 1 ; b d gˆff v i d 1 ; f v i 1 d 1 g and Aso, je i; i 1 jˆjf v i d 1 f v i 1 d 1 j ˆ Xd 1 b j 1 b j V d 1 : b d b U esp C d n U d ; b d 1 b 1 U esp Cn d U d ; b i 1 b i V fo U i U d and so; b d 1 b V d : Then je i; i 1 jˆb d b 1 U d 1 In concusion, d 1 U je i; i 1 j U d 1 fo a i A U: On the othe hand, je i; i 1 j U fo a i B U Let U be the disjoint union of U 1 and U such that ju 1 j V ju j and a e i; i 1 in U 1 (o U ) ae of the sae sign Fo the case ju 1 j > ju j, we have esp T Cn d V je o; j ˆ X 1 e i; i 1 V X je i; i 1 j X X je i; i 1 j je i; i 1 j i A U 1 i A U a contadiction i B U V ju 1 j d 1 ju j d 1 juj ˆ ju 1 j ju j d 1 ju 1 j ju j V d 1 1 > d since > ;

6 300 S Hu et a Fo the case ju 1 jˆju j, say U i ˆfi 1 ; i ; ; i a g fo i ˆ 1; Then je 0; jˆ X 1 e i; i 1 U Xa U Xa e 1 j ;1 j 1 e j ; j 1 X je i; i 1 j i B U d 1 d 1 X i B U ˆ a a ˆ < ; a contadiction Theoe 8 If n ˆ d 1 1 with V, then esp T C d n ˆd Poof The theoe foows fo Theoes 6 and 7 and the fact that gcd n; d 1 ˆ1 Note that Theoe is a specia case to the above theoe when n is odd and d ˆ 1 Fo the case whee n V 5 is odd and d ˆ n 3, we have ˆ 1; ˆ, and Cn d is isoophic to the copeent C n of C n Thus, we have the foowing esut Cooay 9 If n V 5 is odd, then esp T C n ˆ n Acnowedgents The authos than a efeee and Daphne De-Fen Liu fo any usefu suggestions on evising the pape Refeences 1 Cozzens, NB, and Robets, FS: T-cooings of gaphs and the channe assignent pobe, Cong Nueantiu 35, 191±08 (198) Giggs, JR, and Liu, DD-F: Channe assignent pobe fo utuay adjacent sites, J Cob Theoy, Se A68, 169±183 (1994) 3 Hae, WK: Fequency assignent: theoy and appications, Poc IEEE 68, 1497± 1514 (1980) 4 Liu, DD-F: Gaph Hooophiss and the Channe Assignent Pobe, PhD Thesis, Depatent of Matheatics, Univesity of South Caoina, Coubia, SC, 1991

7 T-Cooings and T-Edge Spans of Gaphs Liu, DD-F: T-gaphs and the channe assignent pobe, Discete Math 161, 198± 05 (1996) 6 Robets, FS: T-cooings of gaphs: ecent esuts and open pobes, Discete Math 93, 9±45 (1991) 7 Tesan, BA: List T-cooings of gaphs, Discete App Math 45, 77±89 (1993) Received: May 13, 1996 Revised: Decebe 8, 1997