UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA

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1 UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA On d-graceful labelings Anita Pasotti Quaderni Elettronici del Seminario di Geometria Combinatoria 7E (Maggio 0) Dipartimento di Matematica Guido Castelnuovo Ple Aldo Moro, Roma - Italia

2 On d-graceful labelings Anita Pasotti Abstract We introduce a generalization of the well known concept of a graceful labeling Given a graph Γ with e = d m edges, we call d-graceful labeling of Γ an injective function from V (Γ) to the set 0,,,, d(m + ) } such that f(x) f(y) x, y E(Γ)} =,, 3,, d(m + ) } m +, (m + ),, (d )(m + )} In the case of d = and of d = e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively Also, we call d-graceful α-labeling of a bipartite graph Γ a d-graceful labeling of Γ with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one We show that these new concepts allow to obtain cyclic graph decompositions We investigate the existence of d-graceful α-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles and ladders Introduction As usual, we will denote by K v and K m n the complete graph on v vertices and the complete m-partite graph with parts of size n, respectively For any graph Γ we write V (Γ) for the set of its vertices, E(Γ) for the set of its edges, and D(Γ) for the set of its di-edges, namely the set of all ordered pairs (x, y) with x and y adjacent vertices of Γ If E(Γ) = e, we say that Γ has size e Let Γ be a subgraph of a graph K A Γ-decomposition of K is a set of graphs, called blocks, isomorphic to Γ whose edges partition E(K) In the case that K = K v one also speaks of a Γ-system of order v An automorphism of a Γ-decomposition D of K is a bijection on V (K) leaving D invariant A Γ-decomposition of K is said to be cyclic if it admits an automorphism consisting of a single cycle of length V (K) In this case for giving the set B of blocks it is enough to give a complete system of representatives for the orbits of B under the cyclic group The blocks of such a system are usually called base blocks For a survey on graph decompositions we refer to 3 The concept of a graceful labeling of Γ, introduced by A Rosa, is quite related to the existence problem for cyclic Γ-systems A graceful labeling of a graph Γ of size e is an injective map f from V (Γ) to the set of integers 0,,,, e} such that f(x) f(y) x, y E(Γ)} =,,, e} A graph Γ is said to be graceful if there exists a graceful labeling of it In the case that Γ is bipartite and f has the additional property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one, one says that f is an α-labeling Every graceful labeling of a graph Γ of size e gives rise to a cyclic Γ-system of order e + Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti, 9, I-533 Brescia, Italy anitapasotti@ingunibsit

3 but an α-labeling of Γ gives much more; it gives in fact a cyclic Γ-system of order et + for every positive integer t (see ) and other kinds of graph decompositions such as, for instance, a bicyclic Γ-decomposition of K e (see 8) and a cyclic Γ-decomposition of K m e for every odd integer m coprime with e (see 5) For a very rich survey on graceful labelings we refer to 9 Many variations of graceful labelings have been introduced In particular Gnana Jothi 0 defines an odd graceful labeling of a graph Γ of size e as an injective function f : V (Γ) 0,,,, e } such that f(x) f(y) x, y E(Γ)} =, 3, 5,, e } If such a function exists Γ is said to be odd graceful She proved that every graph with an α- labeling is also odd graceful, while the converse, in general, is not true One of the applications of these labelings is that trees of size e, with a suitable odd graceful labeling, can be used to generate cyclic decompositions of the complete bipartite graph K e For results on odd graceful graphs see, 7, 5 Here we propose the following new definition which is, at the same time, a generalization of the concepts of a graceful labeling and of an odd graceful labeling Definition Let Γ be a graph of size e and let d be a divisor of e, say e = d m Ad-graceful labeling of Γ is an injective function f : V (Γ) 0,,,, d(m + ) } such that f(x) f(y) x, y E(Γ)} =,, 3,, d(m + ) } m +, (m + ),, (d )(m + )} If Γ admits a d-graceful labeling we will say that Γ is a d-graceful graph Note that we find again the concepts of a graceful labeling and of an odd graceful labeling in the two extremal cases of d = and d = e, respectively By saying that d is admissible we will mean that it is a divisor of e and hence that it makes sense to investigate the existence of a d-graceful labeling of Γ It is well known that the complete graph K v is graceful if and only if v, see At the moment we are not able to extend this result to d-graceful labelings of the complete graph with d> We recall that a (v, d, k, λ)-relative difference set (RDS) in a group G is a k-subset S of G such that the list of all differences x y with (x, y) an ordered pair of distinct elements of S covers G H exactly λ times, and no element of H at all with H a suitable subgroup of G of order d (see 3 for a survey on this topic) A RDS as above is said to be cyclic if G is such It is immediate to recognize that every d-graceful labeling of K v can be seen as a cyclic (v(v )+d, d, v, )-RDS though the converse is not true in general For instance, the labeling of the vertices of K 5 with the elements of the set S = 0,,, 9, } is -graceful and we see that S is a (0,, 5, )-RDS Now note that S = 0,, 3,, 0} is also a cyclic (0,, 5, )-RDS, but it is evident that the labeling of the vertices of K 5 with the elements of S is not -graceful The very few known results about cyclic RDSs do not allow us to say much more about the d-gracefulness of the complete graph with d> Also α-labelings can be generalized in a similar way Definition A d-graceful α-labeling of a bipartite graph Γ is a d-graceful labeling of Γ having the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one In the next section we will see how these new concepts can be used to obtain some graph decompositions Then we investigate the existence of d-graceful α-labelings for several classes of

4 bipartite graphs In particular, we prove that paths and stars admit a d-graceful α-labeling for any admissible d We also present partial results about d-graceful labelings of cycles and ladders d-graceful labelings and graph decompositions It is known that graceful labelings are related to difference families, see The concept of a relative difference set has been generalized in to that of a relative difference family A further generalization has been introduced in 6 Definition Let Γ be a graph and let d be a divisor of v A(v, d, Γ, )-difference family (DF for short) is a collection F of injective maps from V (Γ) to Z v such that the list F = f(x) f(y) f F; (x, y) D(Γ)} covers Z v v d Z v exactly once while it does not contain any element of v d Z v, where v d Z v denotes the subgroup of Z v of order d If a (v, d, Γ, )-DF consists of a single map f one says that f is a difference graph In 6 it is proved that Theorem If there exists a (v, d, Γ, )-DF then there exists a cyclic Γ-decomposition of K v d d The next easy proposition establishes the link between d-graceful labelings and DFs Proposition 3 A d-graceful labeling of a graph Γ of size e determines a ( d( e d + ), d, Γ, ) - DF Proof Let f : V (Γ) 0,,, d ( e d +) } be a d-graceful labeling of Γ It is obvious that φ : x V (Γ) f(x) Z d( e +) is a ( d( e d d + ), d, Γ, ) -DF By the above considerations we can state Proposition If there exists a d-graceful labeling of a graph Γ of size e then there exists a cyclic Γ-decomposition of K ( e d +) d Example 5 Figure shows a -graceful labeling of C 6 Hence B = (0,, 3, 6,, 7) is a base block of a cyclic C 6 -decomposition of K Figure : A -graceful labeling of C 6 The following result shows that, as in the case of classical graceful labelings, d-graceful α-labelings are more powerful than d-graceful labelings 3

5 Theorem 6 If there exists a d-graceful α-labeling of a graph Γ of size e then there exists a cyclic Γ-decomposition of K ( e d +) dn for any integer n Proof Let f be a d-graceful α-labeling of Γ, so that we have max f(x) < min f(y ) where X and Y are the two parts of Γ It is easy to see that the set of maps f,f,, f n } from V (Γ) to Z dn( e d +) defined by f i (x) =f(x) x X, i =,, n f i (y) =f(y) + (i )(d + e) y Y, i =,, n is a ( dn( e d + ), dn, Γ, ) -DF Hence by Theorem there exists a cyclic Γ-decomposition of K ( e d +) dn for any integer n Example 7 Figure shows a -graceful α-labeling of C 6 such that there exists a cyclic C 6 - decomposition of K n for any integer n, n The set of base blocks of such a decomposition is given by (0, 5 + 8i,, 3 + 8i,, 7 + 8i) i =0,, n } Figure : A -graceful α-labeling of C 6 3 d-graceful α-labelings of paths and stars In this section we will show that paths and stars have a d-graceful α-labeling for any admissible d From now on, given two integers a and b, by a, b we will denote the set of integers x such that a x b As usual we will denote by P e+ the path on e + vertices, having size e We recall that any path admits an α-labeling, see Theorem 3 Given a positive integer e, the path P e+ has a d-graceful α-labeling for any admissible d Proof Let e = d m and let P dm+ =(x x x dm+ ) We distinguish two cases depending on the parity of m Case : m even Denote by O = x,x 3,,x dm+ } and E = x,x,,x dm } the bipartite sets of P dm+ Let f : V (P dm+ ) 0,d(m + ) be defined as follows: f(x i+ )=i for i 0, dm

6 f(x i )= One can check that f(o) = 0, dm dm (d + )m f(e) = +, (d + )m +3, d(m + ) i for i, m d(m + ) i for i m +,m d(m + ) i for i m +, 3m d(m + ) i (d ) (d + )m (d + 3)m + Hence f is injective and max f(o) < min f(e) Now set for i (d + )m +, (d )m +, dm + d(m + ) m,d(m + ) ε i = f(x i ) f(x i ), ρ i = f(x i ) f(x i+ ) for i =,, dm () By a direct calculation, one can see that ε i,ρ i i =,, m } = (d )(m + ) +,d(m + ) ; ε i,ρ i i = m } +,, m = (d )(m + ) +, (d )(m + ) ; ε i,ρ i i = This concludes case (d )m (d )m ε i,ρ i i = +,, dm } (d )m +,, =m +, (m + ) ; } =,m Case : m odd We divide this case in two subcases depending on the parity of d Case a): d odd In this case the bipartite sets are O = x,x 3,,x dm } and E = x,x,, x dm+ } Let f : V (P dm+ ) 0,d(m + ) be defined as follows: f(x i+ )= f(x i )= i for i 0, m i + for i m+, 3m i + for i 3m+, 5m i + d for i (d )m+, dm d(m + ) i for i,m d(m + ) i for i m +, m d(m + ) i for i m +, 3m d(m + ) i d for i (d )m +, dm+ 5

7 In this case it results that f(o) = 0, m m +3 3m +, 3m +5 5m +3 (d )m + d d(m + ),, d(m + ) (d + )(m + ) f(e) =, (d + )(m + ) (d + 3)(m + ) +, (d + 3)(m + ) (d + 5)(m + ) +, d(m + ) m, d(m + ) So, also in this case, f is injective and max f(o) < min f(e) Note that, now, since dm is odd we have to consider ε i = f(x i ) f(x i ), ρ i = f(x i ) f(x i+ ) for i =,, dm, = f(x dm ) f(x dm+ ) ε dm+ By a long and tedious calculation, one can see that ε i,ρ i i =,, m } ρ m+ ρ (d )m+ } ε m+ ε i,ρ i i = m +3,,m } =(d )(m + ) +, d(m + ) ; } =(d )(m + ) +, } (d )m + ε i,ρ i i = +,, (d )m ε i,ρ i i = +,, dm (d )(m + ) ; } (d )m = =m +, (m + ) ; } } =,m ε dm+ This completes the proof of Case a) Case b): d even Let f : V (P dm+ ) 0,d(m + ) be defined as follows: i for i 0, m i + for i m+, 3m i + for i 3m+ f(x i+ )=, 5m i + d for i (d )m+, dm 6

8 f(x i )= d(m + ) i for i,m d(m + ) i for i m +, m d(m + ) i for i m +, 3m d(m + ) i d for i (d )m +, dm Also here, reasoning as in Case a), it is easy to see that f is a d-graceful α-labeling of P dm+ Proposition 3 There exists a cyclic P e+ -decomposition of K ( e d +) dn for any integers e, n and any divisor d of e Proof It is a direct consequence of theorems 6 and 3 In Figure 3 we have the d-graceful α-labelings of P 9 for d =, 3, 6, 9, 8, obtained following the construction given in the proof of Theorem 3 Note that these are all the possible d- graceful α-labelings of P 9, apart from the already known case of a classical α-labeling (d = ), Figure 3: P 9 We recall that the star on e + vertices S e+ is the complete bipartite graph with one part having a single vertex, called the center of the star, and the other part having e vertices, called external vertices It is obvious that any star has an α-labeling Theorem 33 Given a positive integer e, the star S e+ has a d-graceful α-labeling for any admissible d Proof Let e = d m It is obvious that if we label the center of the star with 0 and the external vertices, arbitrarily, with the elements of,, 3,, d(m + ) } m +, (m + ),, (d )(m + )}, we have a d-graceful α-labeling of S e+ 7

9 Applying Theorem 6 we have: Proposition 3 There exists a cyclic S e+ -decomposition of K ( e d +) dn for any integers e, n and any divisor d of e d-graceful α-labelings of cycles and ladders As usual, we will denote by C k the cycle on k vertices It is obvious that C k is a graph of size k and that it is bipartite if and only if k is even In Rosa proved that C k has an α-labeling if and only if k 0(mod ) Theorem The cycle C k has a -graceful α-labeling for every k Proof Let C k =(x,x,, x k ) and denote by O = x,x 3,, x k } and E = x,x,, x k } its bipartite sets of vertices Consider the map: f : V (C k ) 0,,, k +} defined as follows: We have f(x i+ )=i for i 0, k k + i for i,k f(x i )= k i for i k +, k f(o) = 0, k f(e) = k, 3k 3k +, k + In this way we see that f is injective Also, note that max f(o) < min f(e) Now set ε i,ρ i (for i =,,k) as in (), where the indices are understood modulo k One can easily check that ε i,ρ i i =,, k} = k +, k + and Hence f is a -graceful α-labeling of C k ε i,ρ i i = k +,, k} =, k Theorem The cycle C k admits a -graceful α-labeling for every k Proof Let O and E be defined as in the proof of the previous theorem We are able to prove the existence of a -graceful α-labeling of C k by means of two direct constructions where we distinguish the two cases: k even and k odd Case : k even Consider the map: f : V (C k ) 0,,, k +3} defined as follows: i for i 0, 3k f(x i+ )= i + for i 3k, k k + i for i, k f(x i )= k +3 i for i k +,k k + i for i k +, k 8

10 It is easy to see that f(o) = 0, 3k f(e) = k +, 3k 3k +, k 3k +3, 7k + 7k +, k +3 Hence f is injective and max f(o) < min f(e) Set ε i and ρ i (for i =,, k) as in (), where the indices are understood modulo k It is not hard to see that ε i,ρ i i =,, k } = 3k +, k + 3; ε i,ρ i i = k } +,, k = k +3, 3k + ; ε i,ρ i i = k +,, 3k } =k +3, k; ε i,ρ i i = 3k },, k =,k k +} k +} The thesis follows Case : k odd Let now f : V (C k ) 0,,, k +3} be defined as follows: i for i 0, k f(x i+ )= i + for i k+, k k + i f(x i )= k + i One can check that f(o) = 0, k f(e) = k +, for i,k for i k +, 3k k + i for i 3k+, k k +3 5k +, k 5k +5, 3k + 3k +, k + 3 Also in this case f is injective and max f(o) < min f(e) By a direct calculation, one can see that ε i,ρ i i =,, k } = 3k +5, k + 3; ε i,ρ i i = k + },, k = k +3, 3k + 3k +}; ε i,ρ i i = k +,, 3k } =k +, k; } 3k + ε i,ρ i i =,,k =,k k +} 9

11 The assertion then follows The two figures in Figure show the -graceful α-labeling and the -graceful α-labeling of C 6 provided by the previous theorems Figure : C 6 By virtue of Theorems 6, and, we have Proposition 3 There exists a cyclic C k -decomposition of K (k+) n, and a cyclic C k - decomposition of K (k+) 8n for any integers k, n Theorem For any odd integer k>, the cycle C k is -graceful Proof For k =3, 5, 7 we show directly a -graceful labeling: C 6 = (0, 5,, 3,, 7), C 0 = (0,,, 3, 7,, 5, 0,, 9), C = (0, 5,,,,, 0, 5, 7, 6,, 3, 3, ) Let now k be greater than 7 and set C k =(x,x,x 3,, x k ) Let O = x,x 3,, x k } and E = x,x,, x k } We will construct a -graceful labeling of C k where we distinguish two cases according to whether k (mod ) or k 3(mod ) Case : k (mod ) In this case k =t+ with t even Consider the map f : V (C k ) 0,,, k +} defined as follows: i for i 0, t 7 t +3 i for i t +,t f(x i+ )= 3t + i for i t +,t+ t t + + i for i t + t +, 3 t i t for i 3 t +, t t + i for i, t t + i for i t +,t+ f(x i )= t + + i for i t +,t+ t+ 3t + i for i t + t+ +, 3 t 5t + i for i 3 t +, t + If t = skip the third assignment of f(x i ) One can directly check that f(o) = 0, t 5 t t +3, 3t + t +, t + 0

12 f(e) = t t +, t +3 3 t +, t +, 5 t t + t t +, 3 t + t + +,t t +3, t + + 3t +3, 7 t +3 t + So f is an injective function Now set ε i and ρ i (for i =,, k) as in () where the indices are understood modulo k By a direct calculation one can see that ε i,ρ i i =,, t } = 3t +, t + 3 ε t + = t + } ρ t + ε i,ρ i i = t } +,, t ε t+ } =t +, t + ρ t+ = t We write the remaining differences and distinguish two subcases depending on the parity of t If t is even, it results: ε i,ρ i i = t +,, 5 } t =, t ε 5 t+ = ρ 5 t+ } ρ 3 t+ } ρ t+ =3t +3 ε i,ρ i i = 5 t +,, 3 t } t ε3 t+ } = +,t+ ε i,ρ i i = 3 t +,, t } ε t+ } = t +3, 3t + If t is odd, we have: } 5 } ε i,ρ i i = t +,, t ε 5 t + =, t ρ 5 5 ε i,ρ i i = t +,, 3 } t t ε3 t+} = +,t+ } ρ 3 t+ ε i,ρ i i = 3 } t +,, t ε t+ } = t +3, 3t + t + = ρ t+ =3t +3 This concludes the proof of case Case : k 3(mod ) In this case k =t + with t 5 odd We define f : V (C k ) 0,,, k +} in the following

13 way f(x i+ )= f(x i )= for i 0, t i 7t+5 i for i t+,t 3t + i for i t +,t+ t+ t + + i for i t + t+ 3t +, i t for i 3t+, t t + i for i, t+ t + i for i t+3,t+ t + + i for i t +,t+ t+3 3t + i for i t + t+3 3t+ +, 5t + i for i 3t+3, t + Also in this case one can directly check that f(o) = 0, t 5t +5, 3t + t + 5t +3 t + +, 7t +7 3t + f(e) =, t +3 t +, 3t +3 t +3, t + t + t + t +,t t +3, t + + 3t +3, 7t +5, t + t +3 By long calculations one can verify that, also in case, f is a -labeling of C k In Figure 5 we have the -graceful labeling of C 6 and that of C 30 obtained through the construction given in the proof of Theorem Figure 5: C 6 and C 30

14 To conclude we consider d-graceful α-labelings for ladders We recall that the ladder graph of order k, denoted by L k, can be seen as the cartesian product of the path P = (0 ) of length by the path P k = (0 k ) of length k The ladder graph L k is clearly bipartite with bipartite sets A 0, A where A h is the set of pairs (i, j) 0, } 0,,, k } such that i + j has the same parity as h Theorem 5 The ladder graph L k has a -graceful α-labeling if and only if k is even Proof Since L k has e =3k edges, divides e if and only if k is even Suppose now k to be even One can check that a -graceful α-labeling of L k is given by j if (i, j) A 0 f(i, j) = 3k j if (i, j) A,j=0,, k 3k j if (i, j) A,j= k,,k We observe that the -graceful α-labeling constructed in the previous theorem is very similar to the α-labeling of L k proposed by the author in Figure 6 shows the -graceful α-labeling of L 6 provided by Theorem Figure 6: L 6 As an immediate consequence of Theorem 6 we have Proposition 6 Let k be an even integer There exists a cyclic L k -decomposition of K 3k n for any integer n References RJR Abel and M Buratti, Difference families, in: CRC Handbook of Combinatorial Designs (CJ Colbourn and JH Dinitz eds), CRC Press, Boca Raton, FL (006), C Barrientos, Odd-graceful labelings of trees of diameter 5, ACKE J Graphs Combin 6 (009), D Bryant and S El-Zanati, Graph decompositions, in: CRC Handbook of Combinatorial Designs (CJ Colbourn and JH Dinitz eds), CRC Press, Boca Raton, FL (006),

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