A note on the upper bound and girth pair of (k;g) -cages

Discrete Applied Mathematics 161 (2013) 853–857

Contents lists available at SciVerse ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

Note

A note on the upper bound and girth pair of (k; g )-cages C. Balbuena a , D. González-Moreno b,∗ , J.J. Montellano-Ballesteros c a

Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Spain

b

Departamento de Matemáticas Aplicadas y Sistemas, Universidad Autónoma Metropolitana - Cuajimalpa, Mexico

c

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico

article

info

Article history: Received 29 February 2012 Received in revised form 20 September 2012 Accepted 7 October 2012 Available online 30 October 2012

abstract A (k; g )-cage is a k-regular graph of girth g with minimum order. In this work, for all k ≥ 3 and g ≥ 5 odd, we present an upper bound of the order of a (k; g + 1)-cage in terms of the order of a (k; g )-cage, improving a previous result by Sauer of 1967. We also show that every (k; 11)-cage with k ≥ 6 contains a cycle of length 12, supporting a conjecture by Harary and Kovács of 1983. © 2012 Elsevier B.V. All rights reserved.

Keywords: Cage Girth pair Kronecker product

1. Introduction and notation Through this work only undirected simple graphs are considered. Let G be a graph with vertex set V (G) and edge set E (G). For each vertex v of G, let NG (v) and dG (v) denote the set of neighbors and the degree of v in G, respectively. A graph is said to be k-regular if each of its vertices has degree k. The girth g (G) of G is the length of a shortest cycle in G. Given u, v ∈ V (G), the distance between u and v will be denoted as dG (u, v), and a path (resp. walk) starting from u and ending at v will be called a uv -path (resp. uv -walk). Given xy ∈ E (G) and u ∈ V (G), the distance between xy and u is defined as dG (xy, u) = min{dG (x, u), dG (y, u)}. If H is a cycle, path or walk of G, l(H ) will denote the length of H. For a subset S ⊆ V (G), we denote by G[S ] the subgraph of G induced by S and we write G − S for G[V (G) \ S ]. Given two integers k ≥ 2 and g ≥ 3, a (k; g )-graph is a k-regular graph with girth g. Let n(k; g ) denote the smallest order of a (k; g )-graph. A (k; g )-graph with n(k; g ) vertices is called a (k; g )-cage (a cage for short). Cages were introduced by Tutte [20] in 1947 and their existence was first proved by Sachs [18] in 1963. In 1967 Sauer proved the following theorem. Theorem 1.1 ([19]). For every pair of integers k ≥ 3 and g ≥ 3 odd, n(k; g + 1) ≤ 2n(k; g ). In the following theorem we improve the above result.

∗

Corresponding author. E-mail addresses: [email protected] (C. Balbuena), [email protected] (D. González-Moreno), [email protected] (J.J. Montellano-Ballesteros). 0166-218X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2012.10.008

854

C. Balbuena et al. / Discrete Applied Mathematics 161 (2013) 853–857

Theorem 1.2. Let k ≥ 3 and g ≥ 5 odd. Then

   g −3 k(k − 1) 4 − 2    2n(k; g ) − 2   k−2 n(k; g + 1) ≤   g −1   (k − 1) 4 − 1   2n(k; g ) − 4 k−2

if g ≡ 3 (mod 4), if g ≡ 1 (mod 4).

In [13], Harary and Kovács generalize the concept of (k; g )-cages by replacing the girth condition with a girth pair condition (odd girth and even girth). A graph has girth pair (g , h), with g < h, if g is the girth of the graph and h is the smallest length of a cycle of opposite parity with respect to g. A k-regular graph with girth pair (g , h) of minimum order is said to be a (k; g , h)-cage and the order of a (k; g , h)-cage is denoted by n(k; g , h). In that work the authors proved the existence of (k; g , h)-cages with odd girth g, even girth h, and 3 ≤ g < h and they conjectured that there exists a (k; g )-cage having a cycle of length g + 1 for all k ≥ 3 and odd g. In other words they formulated the following conjecture. Conjecture 1 ([13]). For all k ≥ 3 and g ≥ 3 odd, n(k; g , g + 1) = n(k; g ). In [15] it is proved that, for k ≥ 3 and g ≥ 5, there exists a (k; g )-cage with even girth at most g + (g − 3)/2 if g ≡ 1 (mod 4) and g + (g − 5)/2 if g ≡ 3 (mod 4). This result implies Conjecture 1 for all k ≥ 3 and g = 5, 7. In [4] a lower bound for the order of a k-regular graph with odd girth g ≥ 5 and even girth h ≥ g + 3 is presented. As an application it is also showed in [4] that every (k; g )-cage with k ≥ 3 and g = 5, 7 contains a cycle of length g + 1. In this paper, we also give a lower bound on n(k; g , h) and prove that every (k; 11)-cage contains a cycle of length 12 as corollaries of some other previous results, thus proving Conjecture 1 for g = 11 and k ≥ 6. Corollary 1.3. Let g ≥ 3 be an odd integer and h ≥ g + 1 even. Let 2t = min{h, 2g }. Then n(k; g + 1) ≤ n(k; 2t ) ≤ 2n(k; g , h). Corollary 1.4. For k ≥ 6, every (k; 11)-cage contains a cycle of length 12. For more references on cages, see the survey of Wong [22] or the survey of Exoo and Jajcay [10]. 2. Preliminaries and previous results Let G and H be two graphs. The Kronecker product of G and H, denoted as G ⊗ H, is the graph with vertex set V (G ⊗ H ) = V (G) × V (H ) and edge set E (G ⊗ H ) = {(u, v)(u′ , v ′ ) : uu′ ∈ E (G)? and vv ′ ∈ E (H )}. Observe that |V (G ⊗ H )| = |V (G)| · |V (H )|; |E (G ⊗ H )| = 2 · |E (G)| · |E (H )| and, for every (u, v) ∈ V (G ⊗ H ), its degree is dG⊗H ((u, v)) = dG (u) · dH (v). This product (which is commutative and associative up to isomorphism) is variously known as direct product [6], categorical product [17], tensor product [8] and graph conjunction [5]. It is considered to be one of the most important of all graph products. Several applications and characteristics appear in [7,8,16,17,21]. The following theorem states certain relevant characteristics of the Kronecker product useful to our purposes. Proposition 2.1. Let G be a connected graph with girth g. Then (i) (ii) (iii) (iv)

[21] G ⊗ K2 is a bipartite graph. Furthermore, G ⊗ K2 is disconnected if and only if G is bipartite. [1] For every (u, i), (v, j) ∈ V (G ⊗ K2 ), dG⊗K2 ((u, i), (v, j)) ≥ dG (u, v). [14,16] For every u ∈ V (G), dG⊗K2 ((u, i), (u, j)) ≥ g for i ̸= j. [14,19] Let G be a graph with odd girth g. Then g (G ⊗ K2 ) ≥ g + 1.

Notice also, that the original result of Sauer can be proved by simply taking the Kronecker product G ⊗ K2 where G is

(k; g )-cage with g ≥ 3 odd.

Another key point for proving our results is the so-called Girth Monotonicity Theorem, established by Erdös and Sachs [9], and also by Holton and Sheehan [14], and Fu et al. [11]. Theorem 2.2 ([9,14,11]). Let k ≥ 2 and 3 ≤ g1 < g2 be integers. Then n(k; g1 ) < n(k; g2 ).

C. Balbuena et al. / Discrete Applied Mathematics 161 (2013) 853–857

855

Fig. 1. k = 3 and g = 9.

We also need the following results by Gács and Héger [12] and Araujo et al. [2,3] and Theorem 2.3. Let k ≥ 3 be an integer and let q be the smallest prime power such that k ≤ q. Then (i) [12] n(k; 12) ≤ 2(k5 − k3 ); 2 2 (ii) [2] n(k; 12) ≤ 2kq  (q − 1); (iii) [3] n(k; 12) ≤

 2k(k − 1)4  2k(k − 1)4

 4 7

if 7 ≤ k ≤ 3275,

6

 1+

4

1 2 ln2 (k − 1)

if k ≥ 3276.

3. Proofs of the main results Proof of Theorem 1.2. Let G be a (k; g )-cage with k ≥ 3 and g ≥ 5 odd. Case 1. g ≡ 3 (mod 4). g −3 g +1 Let v0 ∈ V (G), A = {z ∈ V (G) : dG (v0 , z ) ≤ 4 } and X = {z ∈ V (G) : dG (v0 , z ) = 4 }. Let G′ = G − A. Observe that X ⊆ V (G′ ). Therefore the following two properties are satisfied: (a) For every u ∈ V (G′ ) \ X , dG′ (u) = dG (u) and for every u ∈ X , dG′ (u) = dG (u) − 1. (b) For every u, v ∈ X , dG′ (u, v) ≥ g − (dG (v0 , u) + dG (v0 , v)) = g − 2(

g +1 4

)=

g −1 . 2

Since g is odd, by Proposition 2.1, it follows that G′ ⊗ K2 is bipartite and g (G′ ⊗ K2 ) ≥ g + 1. Suppose V (K2 ) = {0, 1} and let H be the graph obtained by adding to G′ ⊗ K2 the set of edges E ′ = {(u, 0)(u, 1) : u ∈ X }. By (a), H is a k-regular graph. Let C be a cycle in H. If E (C ) ∩ E ′ = ∅, then C is totally contained in G′ ⊗ K2 and therefore l(C ) ≥ g (G′ ⊗ K2 ) ≥ g + 1. If E (C ) ∩ E ′ = {(u, 0)(u, 1)} for some u ∈ X , then C contains an (u, 0)(u, 1)-path totally contained in G′ ⊗ K2 . From Proposition 2.1(iii) it follows that l(C ) ≥ dG′ ⊗K2 ((u, 0), (u, 1)) + 1 ≥ g + 1. If |E (C ) ∩ E ′ | ≥ 2 then C contains at least two paths, P1 and P2 , between vertices of X totally contained in G′ ⊗ K2 . Then by Proposition 2.1(i) and property (b) it follows that l(C ) ≥ 2 + l(P1 ) + l(P2 ) ≥ 2 +

g −1 2

+

g −1 2

= g + 1.

Therefore H is a (k; g (H ))-graph with g (H ) ≥ g + 1 and |V (H )| ≥ n(k; g (H )). Hence, by Theorem 2.2, n(k; g + 1) ≤ |V (H )| = 2|V (G′ )| = 2(|V (G)| − |A|) = 2(n(k; g ) − |A|). A simple counting argument on the number of vertices at distance at most

g −3 2

from vertex v0 in G gives

 g −7  g −3 4  k(k − 1) 4 − 2 i |A| = 1 + k  (k − 1)  = k−2 i=0 and the first case of the theorem holds. Case 2. g ≡ 1 (mod 4). g −5 g −1 Let v0 ∈ V (G), and w0 ∈ NG (v0 ). Let A = {z ∈ V (G) : dG (v0 , z ) ≤ 4 } and B = {z ∈ V (G) : dG (v0 , z ) = 4 } ∩ {z ∈

V (G) : dG (w0 , z ) =

g −5 4

} (see Fig. 1).

Let X = {z ∈ V (G) : dG (v0 , z ) =

g −1 4

} \ B and Y = {z ∈ V (G) : dG (v0 , z ) =

g +3 4

} ∩ {z ∈ V (G) : dG (w0 , z ) =

g −1 4

}.

856

C. Balbuena et al. / Discrete Applied Mathematics 161 (2013) 853–857

Fig. 2. Petersen graph P and the (3; 6)-graph on 16 vertices isomorphic to K2 ⊗ (P − {9, 10}) ∪ {(2, 0)(2, 1), (4, 0)(4, 1), (6, 0)(6, 1), (8, 0)(8, 1)}.

Let G′ = G − (A ∪ B). Observe that (X ∪ Y ) ⊆ V (G′ ) and, as in the previous case, the following properties are satisfied: (a) If u ∈ V (G′ ) \ (X ∪ Y ) then dG′ (u) = dG (u) and if u ∈ X ∪ Y then dG′ (u) = dG (u) − 1. (b) For every u, v ∈ (X ∪ Y ) it follows that: • if u, v ∈ X , dG′ (u, v) ≥ g − (dG (v0 , u) + dG (v0 , v)) ≥ g − 2( g −4 1 ) = g +2 1 ;

• if u, v ∈ Y , dG′ (u, v) ≥ g − (dG (w0 , u) + dG (w0 , v)) ≥ g − 2( g −4 1 ) = g +2 1 ; • if u ∈ X and v ∈ Y , dG′ (u, v) ≥ g − (dG (v0 , u) + dG (v0 , v)) ≥ g − ( g −4 1 + g +4 3 ) =

g −1 . 2

Suppose V (K2 ) = {0, 1} and let H be the graph obtained by adding to G′ ⊗ K2 the set of edges E ′ = {(u, 0)(u, 1) : u ∈ X ∪ Y }. From here, reasoning in an analogous way as in the previous case, we obtain that H is a (k; g + 1)-graph whose order satisfies that n(k; g + 1) ≤ |V (H )| = 2|V (G′ )| = 2(|V (G)| − (|A| + |B|)) = 2(n(k; g ) − (|A| + |B|)). Now just observe that |A| = follows.

g −5 k(k−1) 4 −2 k−2

and |B| = (k − 1)

g −5 4

 . Therefore |A| + |B| = 2

g −1

(k−1) 4 −1 k−2

 and the result



Remark 3.1. Note that the proof of Theorem 1.2 provides us with a method for constructing (k; g + 1)-graphs from (k; g )cages of odd girth. These graphs have order equal to the upper bound. Fig. 2 depicts Petersen graph and the corresponding (3; 6)-graph obtained from it on 16 vertices, which is just two vertices bigger than the (3; 6)-cage. Proof of Corollary 1.3. Taking into account that if the girth g of G is odd, then G ⊗ K2 is a bipartite graph in which C2t ⊗ K2 ∼ = C2t ∪ C2t for all even cycles C2t of G, and C2t +1 ⊗ K2 ∼ = C4t +2 for all odd cycles C2t +1 of G, t ≥ 1. Therefore, G ⊗ K2 is a bipartite graph with girth equal to 2t = min{2g , h}. Hence n(k; 2t ) ≤ 2n(k; g , h). Using Theorem 2.2, the result n(k; g + 1) ≤ n(k; 2t ) ≤ 2n(k; g , h) is immediate.  Proof of Corollary 1.4. Let G be a (k; 11)-cage, with k ≥ 6, and let h(G) ≥ 12 be its even girth, that is, G is a (k; 11, h(G))h(G)−2 cage, and therefore n(k; 11) = n(k; 11, h(G)). On the one hand, counting the number of vertices at distance at most 2 from an edge in G, we obtain that n(k; h(G)) ≥

2(k − 1)

h(G) 2

k−2

−2

.

(1)

By Corollary 1.3, and Theorem 2.2, it follows that n(k; h(G)) ≤ 2n(k; 11) < 2n(k; 12). Thus, by Theorem 2.3 and (1), we have

 5 4(k − k3 )     4   h(G) 7  2(k − 1) 2 − 2 4k(k − 1)4 < 6  k−2      4k(k − 1)4 1 +

if k = 6, if 7 ≤ k ≤ 3275, 1 2ln2 (k − 1)

4 if k ≥ 3276;

C. Balbuena et al. / Discrete Applied Mathematics 161 (2013) 853–857

857

where q is the smallest prime power such that k ≤ q. It is not difficult to see that if h(G) ≥ 14 the above inequality is not true for k ≥ 4. Hence we conclude that h(G) ≤ 12 and the result follows.  The Balaban (3; 11)-cage is unique and can be constructed by excision from the (3; 12)-cage. Since it contains 12-cycles, in order to prove Conjecture 1 for g = 11 it is only necessary to check the cases k = 4, 5. Acknowledgments We wish to thank the anonymous referees for their valuable suggestions. The research was supported by the Ministerio de Educación y Ciencia, Spain, the European Regional Development Fund (ERDF) under project MTM2011-28800-C02-02, under the Catalonian Government project 1298 SGR2009 and by PROMEP under project 47510249. References [1] G. Abay-Asmerom, R. Hammack, Centers of tensor products of graphs, Ars Combin. 74 (2005) 201–211. [2] G. Araujo-Pardo, C. Balbuena, T. Héger, Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages, Discrete Math. 310 (8) (2010) 1301–1306. [3] G. Araujo-Pardo, D. González, J. Montellano-Ballesteros, O. Serra, On upper bounds and connectivity of cages, Australas. J. Combin. 38 (2007) 221–228. [4] C. Balbuena, T. Jiang, Y. Lin, X. Marcote, M. Miller, A lower bound on the order of regular graphs with given girth pair, J. Graph Theory 55 (2) (2007) 155–163. [5] J.-C. Bermond, Hamiltonian decompositions of graphs, digraphs and hypergraphs, Ann. Discrete Math. 3 (1978) 21–28. [6] J. Bosak, Decompositions of Graphs, Kluwer Academic, Dordrecht, 1991. [7] A. Bottreau, Y. Métivier, Some remarks on the Kronecker product of graphs, Inform. Process. Lett. 68 (1998) 55–61. [8] M.F. Capobianco, On characterizing tensor–composite graphs, Ann. New York Acad. Sci. 175 (1970) 80–84. [9] P. Erdös, H. Sachs, Regulare Graphen gegebener Taillenweite mit minimaler Knotenzahl, Wiss. Z. Martin-Luther-Univ. Halle Wittwnber Math-Natur. Reih. 12 (1963) 251–258. [10] G. Exoo, R. Jajcay, Dynamic cage survey, Electron. J. Combin. 15 (2008). [11] H. Fu, K. Huang, C. Rodger, Connectivity of cages, J. Graph Theory 24 (1997) 187–191. [12] A. Gács, T. Héger, On geometric constructions of (k, g )-graphs, Contrib. Discrete Math. 3 (2008) 63–80. [13] F. Harary, P. Kovács, Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209–218. [14] D.A. Holton, J. Sheehan, The Petersen Graph, in: Australian Mathematical Society Lecture Notes, vol. 7, Cambridge University Press, Cambridge, 1993. [15] T. Jiang, Short even cycles in cages with odd girth, Ars Combin. 59 (2001) 165–169. [16] S.-R. Kim, Centers of a tensor composite graph, in: Proceedings of the 22nd Southeastern Conference on Combinatorics, Graph Theory, and Computing, Baton Rouge, LA, 1991, in: Congr. Numer., vol. 81, 1991, pp. 193–203. [17] D.J. Miller, The categorical product of graphs, Canad. J. Math. 20 (1968) 1511–1521. [18] H. Sachs, Regular graphs with given girth and restricted circuits, J. Lond. Math. Soc. 38 (1963) 423–429. [19] N. Sauer, Extremaleigenschaften regulärer Graphen gegebener Taillenweite, I and II, Sitzber. Österreich. Acad. Wiss. Math. Natur. Kl., S-B II 176 (1967) 9–25. 176 (1967), 27–43. [20] W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. (1947) 459–474. [21] P.M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962) 47–52. [22] P.K. Wong, Cages—a survey, J. Graph Theory 6 (1982) 1–22.