42 RICHARD M. LOW AND SIN-MIN LEE grphs nd Z k mgic grphs were investigted in [4,7,8,1]. The construction of mgic grphs ws studied by Sun nd Lee [18].

Transcription

1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 34 (26), Pges On the products of group-mgic grphs Richrd M. Low Λ Deprtment of Mthemtics Sn Jose Stte University Sn Jose, CA U.S.A. Sin-Min Lee Λ Deprtment of Computer Science Sn Jose Stte University Sn Jose, CA U.S.A. This pper is dedicted to the memory of Gwong C. Sun. Abstrct Let A be n belin group. We cll grph G =(V; E) A mgic if there exists lbeling f : E(G)! A fg such tht the induced vertex set lbeling f + : V (G)! A, defined by f + (v) =±f(u; v) where the sum is over ll (u; v) 2 E(G), is constnt mp.for four clssicl products, we exmine the A mgic property of the resulting grph obtined from the product of two A mgic grphs. 1 Introduction Let G be connected grph without multiple edges or loops. For ny belin group A (written dditively), let A Λ = A fg. A function f : E(G)! A Λ is clled lbeling of G. Any such lbeling induces mp f + : V (G)! A, defined by f + (v) =±f(u; v) where the sum is over ll (u; v) 2 E(G). If there exists lbeling f whose induced mp on V (G) is constnt mp, we sythtf is n A mgic lbeling nd tht G is n A mgic grph. Z mgic grphs were considered by Stnley [16,17], where he pointed out tht the theory of mgic lbelings could be studied in the generl context of liner homogeneous diophntine equtions. Doob [1,2,3] nd others [6,9,11] hve studied A mgic Λ The uthors wish to thnk To-Ming Wng for his helpful comments

2 42 RICHARD M. LOW AND SIN-MIN LEE grphs nd Z k mgic grphs were investigted in [4,7,8,1]. The construction of mgic grphs ws studied by Sun nd Lee [18]. In this pper, we extend some results to A mgic grphs. In prticulr, grph products offer stright forwrd nd systemtic mens of constructing A mgic grphs. Within the mthemticl literture, vrious definitions of mgic grphs hve been introduced. The originl concept of n A mgic grph is due to J. Sedlcek [14,15], who defined it to be grph with rel vlued edge lbeling such tht (i) distinct edges hve distinct nonnegtive lbels, nd (ii) the sum of the lbels of the edges incident to prticulr vertex is the sme for ll vertices. Previously, Kotzig nd Ros [5] hd introduced yet nother definition of mgic grph. Over the yers, there hs been gret reserch interest in grph lbeling problems. The interested reder is directed to Wllis' [19] recent monogrph on mgic grphs. 2 Definitions nd exmples For ny commuttive ringr with unity, U(R) denotes the multiplictive group of units in R. The product of two grphs G 1 (p 1 ;q 1 )=(V 1 ;E 1 )ndg 2 (p 2 ;q 2 )=(V 2 ;E 2 ) cn be defined in vrious wys. Within the product, the vertices will be denoted by (; b) : 2 V 1 nd b 2 V 2, nd the edges will be denoted by ((; b); ( ;b )) : ; 2 V 1 nd b; b 2 V 2. Let us recll the following definitions of vrious products of grphs. Definition 1. Crtesin product G 1 G 2 : V (G 1 G 2 )=V 1 V 2 = f(; b) : 2 V 1 ^ b 2 V 2 g nd E(G 1 G 2 )=f((; b); ( ;b )) : ; 2 V 1 ^ b; b 2 V 2 ^ (( = ^ (b; b ) 2 E 2 ) _ (b = b ^ (; ) 2 E 1 ))g. Definition 2. Lexicogrphic product G 1 ffl G 2 : V (G 1 ffl G 2 )=V 1 V 2 = f(; b) : 2 V 1 ^ b 2 V 2 g nd E(G 1 ffl G 2 )=f((; b); ( ;b )) : ; 2 V 1 ^ b; b 2 V 2 ^ (( = ^ (b; b ) 2 E 2 ) _ (; ) 2 E 1 )g. Definition 3. Tensor product G 1 Ω G 2 : V (G 1 Ω G 2 ) = V 1 V 2 = f(; b) : 2 V 1 ^ b 2 V 2 g nd E(G 1 Ω G 2 )=f((; b); ( ;b )) : ; 2 V 1 ^ b; b 2 V 2 ^ (; ) 2 E 1 ^ (b; b ) 2 E 2 g. Definition 4. Norml product G 1?G 2 : V (G 1?G 2 ) = V 1 V 2 = f(; b) : 2 V 1 ^ b 2 V 2 g nd E(G 1?G 2 )=E(G 1 G 2 ) [ E(G 1 Ω G 2 ), where E(G 1 G 2 )nd E(G 1 Ω G 2 ) re the edge sets of the Crtesin nd conjunctive products of G 1 nd G 2 respectively. The tensor product (lso clled the Kronecker product [2], ctegoricl product [12] nd conjunctive product) is one of the lest understood grph products. The lexicogrphic product is lso known s composition nd ws introduced by Sbidussi [13]. Note tht of the four products, only the lexicogrphic product is not commuttive. We conclude this section by giving few exmples where the product of two grphs is A mgic, but the individul fctors re not A mgic. Exmple 1. Consider the grph G = P 4 P 4. Figure 1 shows tht G is Z k mgic, for k 6= 2. However, P 4 is not Z k mgic, for ny k.

3 ON THE PRODUCTS OF GROUP-MAGIC GRAPHS 43 Figure 1. G = P 4 P 4. Exmple 2. Consider the grph G = P 4 ffl N 2, where N 2 is the null grph of order two (Figure 2). Since G is n eulerin grph with n even number of edges, we cn lbel the edges of the eulerin circuit with ; ; ; ; :::; ;, where 2 A Λ. Thus, G is A mgic. Clerly, P 4 nd N 2 re not A mgic. Figure 2. G = P 4 ffl N 2. Exmple 3. Consider the grph G = P 4 ΩP 4. Figure 3 shows tht G is Z 2k+1 mgic, for ll k. Clerly, P 4 is not Z 2k+1 mgic. Figure 3. G = P 4 Ω P 4. 3 Products of group mgic grphs Let us nownlyze the A mgic property of the resulting grph obtined from the product of two A mgic grphs. For A mgic grphs G 1 (p 1 ;q 1 )ndg 2 (p 2 ;q 2 ), let L 1 nd L 2 represent their respective A mgic lbelings. Furthermore, let w 1 nd w 2 be the constnts induced on V 1 nd V 2 respectively, by these lbelings. Thus, we hve P L 1(; )=w 1 for ny vertex 2 V 1 nd P b L 2(b; b )=w 2 for ny vertex b 2 V 2.

4 44 RICHARD M. LOW AND SIN-MIN LEE To illustrte the theorems in this section, we will use the lbeled grphs found in Figure 4. Figure 4. Z 7 mgic lbelings of G 1 nd G 2. Theorem 1. Let A be n belin group. If G 1 = (V 1 ;E 1 ) nd G 2 = (V 2 ;E 2 ) re A mgic grphs, then the Crtesin product G 1 G 2 is A mgic. Proof. Let L denote the lbeling ssignment ofe(g 1 G 2 ), defined by: ( L 1 (; ); if b = b : L((; b); ( ;b )) = L 2 (b; b ); if = : Then, the induced lbeling of every vertex (; b) is: ;b L((; b); ( ;b )) = b L((; b); (; b )) + L((; b); ( ;b)) = b = w 2 + w 1 : L 2 (b; b )+ L 1 (; ) Figure 5. Z 7 mgic lbeling of the Crtesin product G 1 G 2. Theorem 2. Let A be nbelin group. If G 1 =(V 1 ;E 1 ) nd G 2 =(V 2 ;E 2 ) re A mgic grphs, then both the lexicogrphic products G 1 ffl G 2 nd G 2 ffl G 1 re A mgic.

5 ON THE PRODUCTS OF GROUP-MAGIC GRAPHS 45 Proof. We will showtht G 1 ffl G 2 is A mgic. Let L denote the lbeling ssignment of E(G 1 ffl G 2 ), defined by: ( L 2 (b; b ); if = : L((; b); ( ;b )) = L 1 (; ); otherwise: Then, the induced lbeling of every vertex (; b) is: ;b L((; b); ( ;b )) = ;b = = b = w 2 + L((; b); ( ;b )) + ;b L 2 (b; b )+ fp 2 L 1 (; )g 6= b L 1 (; ) L((; b); ( ;b )) = w 2 + p 2 w 1 : A similr rgument is used to showtht G 2 ffl G 1 is A mgic. Figure 6. Z 7 mgic lbeling of the lexicogrphic product G 1 ffl G 2. Theorem 3. Let A be nbelin group, underlying commuttive ring R. If there exist A mgic lbelings L 1 : E(G 1 )! A Λ U(R) nd L 2 : E(G 2 )! A Λ U(R) for grphs G 1 nd G 2 respectively, then the tensor product G 1 Ω G 2 is A mgic. Proof. Let L denote the lbeling ssignment ofe(g 1 Ω G 2 ), defined by: L((; b); ( ;b )) = L 1 (; ) L 2 (b; b ): Then, the induced lbeling of every vertex (; b) is:

6 46 RICHARD M. LOW AND SIN-MIN LEE ;b L((; b); ( ;b )) = = = w 1 w 2 : fl 1 (; ) L 2 (b; b )g b L 1 (; ) L 2 (b; b ) b Note tht L ssigns non zero elements to E(G 1 Ω G 2 ), since the rnge of L 1 nd L 2 re subsets of A Λ U(R). Corollry 1. Let A be nbelin group, underlying field F. If G 1 =(V 1 ;E 1 ) nd G 2 =(V 2 ;E 2 ) re A mgic grphs, then the tensor product G 1 Ω G 2 is A mgic. Proof. This is n immedite consequence of Theorem 3. Figure 7. Z 7 mgic lbeling of the tensor product G 1 Ω G 2. Theorem 4. Let A be nbelin group, underlying commuttive ring R. If there exist A mgic lbelings L 1 : E(G 1 )! A Λ U(R) nd L 2 : E(G 2 )! A Λ U(R) for grphs G 1 nd G 2 respectively, then the norml product G 1?G 2 is A mgic. Proof. Note the following: E(G 1?G 2 ) = E(G 1 G 2 ) [ E(G 1 Ω G 2 ) nd E(G 1 G 2 ) E(G 1 Ω G 2 )=;. Let L denote the lbeling ssignment ofe(g 1?G 2 ), defined by: L((; b); ( ;b )) = 8 >< >: L 1 (; ); if b = b : L 2 (b; b ); if = : L 1 (; ) L 2 (b; b ); otherwise: Then, the induced lbeling of every vertex (; b) is:

7 ON THE PRODUCTS OF GROUP-MAGIC GRAPHS 47 L((; b); ( ;b )) = L((; b); (; b )) + L((; b); ( ;b)) ;b b + 6= = b + b b 6=b L((; b); ( ;b )) L 2 (b; b )+ L 1 (; ) = w 2 + w 1 + w 1 w 2 : L 1 (; ) b L 2 (b; b ) L ssigns non zero elements to E(G 1?G 2 ), since the rnge of L 1 nd L 2 re subsets of A Λ U(R). Corollry 2. Let A be nbelin group, underlying field F. If G 1 =(V 1 ;E 1 ) nd G 2 =(V 2 ;E 2 ) re A mgic grphs, then the norml product G 1?G 2 is A mgic. Proof. This is n immedite consequence of Theorem 4. Figure 8. Z 7 mgic lbeling of the norml product G 1?G 2. References [1] M. Doob, On the construction of mgic grphs, Proc. Fifth S.E. Conference on Combintorics, Grph Theory nd Computing (1974), [2] M. Doob, Generliztions of mgic grphs, J. Combin. Theory Ser. B 17 (1974), [3] M. Doob, Chrcteriztions of regulr mgic grphs, J. Combin. Theory Ser. B 25 (1978), [4] M.C. Kong, S-M Lee nd H. Sun, On mgic strength of grphs, Ars Combintori 45 (1997),

8 48 RICHARD M. LOW AND SIN-MIN LEE [5] A. Kotzig nd A. Ros, Mgic vlutions of finite grphs, Cnd. Mth. Bull. 13 (197), [6] S-M Lee, F. Sb, E. Slehi nd H. Sun, On the V 4 group mgic grphs, Congressus Numerntium 156 (22), [7] S-M Lee, F. Sb nd G.C. Sun, Mgic strength of the k-th power of pths, Congressus Numerntium 92 (1993), [8] S-M Lee nd E. Slehi, Integer-mgic spectr of mlgmtions of strs nd cycles, Ars Combintori 67 (23), [9] S-M Lee, Hugo Sun nd Ixin Wen, On group-mgic grphs, J. Combin. Mth. Combin. Computing 38 (21), [1] S-M Lee, L. Vldes nd Yong-Song Ho, On integer-mgic spectr of trees, double trees nd bbrevited double trees, J. Combin. Mth. Combin. Computing 46 (23), [11] R.M. Low nd S-M Lee, On group-mgic eulerin grphs, J. Combin. Mth. Combin. Computing 5 (24), [12] D.J. Miller, The ctegoricl product of grphs, Cnd. J. Mth. 2 (1968), [13] G. Sbidussi, The lexicogrphic product of grphs, Duke Mth. J. 28 (1961), [14] J. Sedlcek, On mgic grphs, Mth. Slov. 26 (1976), [15] J. Sedlcek, Some properties of mgic grphs, in Grphs, Hypergrph, nd Bloc Syst. 1976, Proc. Symp. Comb. Anl., Zielon Gor (1976), [16] R.P. Stnley, Liner homogeneous diophntine equtions nd mgic lbelings of grphs, Duke Mth. J. 4 (1973), [17] R.P. Stnley, Mgic lbeling of grphs, symmetric mgic squres, systems of prmeters nd Cohen-Mculy rings, Duke Mth. J. 4 (1976), [18] G.W. Sun nd S-M Lee, Construction of Mgic Grphs, Congressus Numerntium 13 (1994), [19] W.D. Wllis, Mgic Grphs, Birkhuser Boston, (21). [2] P.M. Weichsel, The Kronecker product of grphs, Proc. Amer. Mth. Soc. 13 (1962), (Received 8 Apr 24)