Decompositions into generalized Petersengraphs via graceful labelings

Anita Pasotti 1 Anna Benini 2

Dipartimento di Matematica,

Universita di Brescia

Brescia, Italy

Abstract

We prove that the generalized Petersen graph P8n,3 admits an α-labeling for anyinteger n ≥ 1 confirming that the conjecture posed by A. Vietri in [10] is true. Wepresent also a result about the existence of d-divisible α-labelings of P8n,3. In sucha way we obtain cyclic decompositions of the complete graph and of the completemultipartite graph into copies of P8n,3.

Keywords: generalized Petersen graph, α-labeling, graph decomposition.

1 Introduction

As usual, we denote by Kv and Km×n the complete graph on v vertices and thecomplete m-partite graph with parts of size n, respectively. Given a subgraphΓ of a graph K, a Γ-decomposition of K is a set of graphs, called blocks, iso-morphic to Γ, whose edges partition the edge-set of K. Such a decompositionis said to be cyclic when it is invariant under a cyclic permutation of all the

1 Email: anita.pasotti@ing.unibs.it2 Email: anna.benini@ing.unibs.it

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vertices of K.The concept of a graceful labeling of a graph Γ, introduced by A. Rosa [7],is proved to be an useful tool for determining the existence of cyclic Γ-decompositions of the complete graph. A graceful labeling of a graph Γ ofsize e is an injective function f : V (Γ) → {0, 1, 2, . . . , e} such that

{|f(x)− f(y)| | [x, y] ∈ E(Γ)} = {1, 2, . . . , e}.

In the case where Γ is bipartite and f has the additional property that itsmaximum value on one of the two bipartite sets does not reach its minimumon the other one, one says that f is an α-labeling. In [7], Rosa proved thatif a graph Γ of size e admits a graceful labeling f then there exists a cyclicΓ-decomposition of K2e+1 and if f is, in addition, an α-labeling then thereexists a cyclic Γ-decomposition of K2et+1 for any positive integer t.If Γ is the Petersen graph P5,2 the existence of Γ-decompositions of the com-plete graph has been completely solved in [1]. On the contrary, as far as weare aware, when Γ is a generalized Petersen graph Pn,k very little is knownabout the existence of Γ-decompositions of Kv, see [3,8,10].In [2] we made a small progress into this problem proving the existence of acyclic P8n,3-decomposition of K24nt+1 for any pair of positive integers n and t.We obtained this result via the construction of an α-labeling of P8n,3, provingin this way the conjecture posed by A. Vietri in [10].

2 Main result

We recall that, given two positive integers n and k such that n ≥ 3 and1 ≤ k ≤ ⌊n−1

2⌋, the generalized Petersen graph Pn,k is the graph whose vertex

set is {ai, bi : 1 ≤ i ≤ n} and whose edge set is {[ai, bi], [ai, ai+1], [bi, bi+k] :1 ≤ i ≤ n}, where subscripts are meant modulo n. Clearly, the generalizedPetersen graph Pn,1 is nothing but a prism on 2n vertices.The only results about gracefulness of infinite classes of Pn,k’s other thanprisms (whose gracefulness has been proved in [4], for a more recent proof see[5]) were obtained by Vietri, see [8,9,10]. Also, in [10] the author conjecturesthat there exists an α-labeling for every graph P8n,3.In this paper we explain how we have proved Vietri’s conjecture. In [2], thefollowing theorems are proved and the construction of the α-labeling of P8n,3

can be found.

Theorem 2.1 For any positive integer n ≥ 1, P8n,3 admits an α-labeling.

As an immediate consequence we have:

A. Pasotti, A. Benini / Electronic Notes in Discrete Mathematics 40 (2013) 295–298296

Theorem 2.2 There exists a cyclic P8n,3-decomposition of K24nt+1 for any

positive integers t and n.

3 d-divisible α-labelings

In our paper we present also our work in progress on the existence of d-divisibleα-labelings of P8n,3, concept introduced in [6] as a generalization of gracefullabelings.

Definition 3.1 Let Γ be a graph of size e = d · m. A d-divisible graceful

labeling of Γ is an injective function f : V (Γ) → {0, 1, 2, . . . , d(m + 1) − 1}such that

{|f(x)− f(y)| | [x, y] ∈ E(Γ)} =

= {1, 2, 3, . . . , d(m+ 1)− 1} \ {m+ 1, 2(m+ 1), . . . , (d− 1)(m+ 1)}.

Also, a d-divisible α-labeling of a bipartite graph Γ is a d-divisible gracefullabeling of Γ having the property that its maximum value on one of the twobipartite sets does not reach its minimum value on the other one.

It is immediate that a 1-divisible α-labeling is nothing but a classical α-labeling.Currently, we are considering the following open problem.

Problem 3.2 Determine the set of values of d for which the graph P8n,3 ad-

mits a d-divisible α-labeling for any positive integer n.

Up to now, we have proved the following result.

Theorem 3.3 For any positive integer n ≥ 1, P8n,3 admits a 2-divisible α-

labeling.

In [6] it is proved that if a graph Γ of size e admits a d-divisible gracefullabeling f then there exists a cyclic Γ-decomposition of K( e

d+1)×2d and that if

f is a d-divisible α-labeling of Γ then there exists a cyclic Γ-decomposition ofK( e

d+1)×2dt for any integer t ≥ 1.

So, the following result is a consequence of Theorem 3.3.

Theorem 3.4 There exists a cyclic P8n,3-decomposition of K(12n+1)×4t for any

positive integers t and n.

A. Pasotti, A. Benini / Electronic Notes in Discrete Mathematics 40 (2013) 295–298 297

References

[1] Adams, P., and D. E. Bryant, The spectrum problem for the Petersen graph, J.Graph Theory 22 (1996), 175–180.

[2] Benini, A., and A. Pasotti, α-labelings of a class of generalized Petersen graphs,submitted.

[3] Bonisoli, A., M. Buratti, and G. Rinaldi, Sharply transitive decompositions of

complete graphs into generalized Petersen graphs, Innov. Incidence Geom. 6/7(2007/08), 95–109.

[4] Frucht, R., and J. A. Gallian, Labeling prisms, Ars Combin. 26 (1988), 69–82.

[5] Pasotti, A., Constructions for cyclic Moebius ladder systems, Discrete Math.310 (2010), 3080–3087.

[6] Pasotti, A., On d-graceful labelings, to appear on Ars Combin.

[7] Rosa, A., On certain valuations of the vertices of a graph, Theory of Graphs(Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and DunodParis (1967), 349–355.

[8] Vietri, A., A new infinite family of graceful generalised Petersen graphs, via

“graceful collages” again, Australas. J. Combin. 41 (2008), 273–282.

[9] Vietri, A., Erratum: A little emendation to the graceful labelling of the

generalised Petersen graph P8t,3 when t = 5: “Graceful labellings for an

infinite class of generalized Petersen graphs” [Ars. Combin. 81 (2006), 247–255; MR2267816], Ars Combin. 83 (2007), 381.

[10] Vietri, A., Graceful labellings for an infinite class of generalised Petersen graphs,Ars Combin. 81 (2006), 247–255.

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