GABRIELA ARAUJO ET AL. 1 Introduction Given two integers k and g 3, a (k; g)-graph is a k-regular graph G with girth g(g) = g. A(k; g)-graph of minimu

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 38 (007), Pages 1 8 On upper bounds and connectivity of cages Gabriela Araujo Λ Diego González Λ Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria Coyoacán Mexico garaujo@math.unam.mx gmoreno@math.unam.mx Juan José Montellano-Ballesteros Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria Coyoacán Mexico juancho@math.unam.mx Oriol Serra y Dep. Matem atica Aplicada IV Universitat Polit ecnica de Catalunya Barcelona Spain oserra@ma4.upc.edu Abstract In this paper we give an upper bound for the order of (k; g)-cages when k 1 is not a prime power and g f6; 8; 1g. As an application we obtain new upper bounds for the order of cages when g =11andg =1 and k 1 is not a prime power. We also confirm a conjecture of Fu, uang and Rodger on the k-connectivity of(k; g)-cages for g = 1, and for g =7; 11 when k 1 is a prime power. Λ Supported by PAPIIT-UNAM, México, Proyecto IN y Supported by the Spanish Research Council under grant MTM C0-01 and the Catalan Research Council under grant 005SGR0056.

2 GABRIELA ARAUJO ET AL. 1 Introduction Given two integers k and g 3, a (k; g)-graph is a k-regular graph G with girth g(g) = g. A(k; g)-graph of minimum order is called a (k; g)-cage. For references on cages see for instance the survey of Wong [9] or the website of Royle [7], which contains the current best known bounds for the order of (k; g) cages. We denote as ν(k; g) the order of the (k; g)-cages. The problem of determining the value of ν(k; g) is still wide open for most pairs (k; g). By counting the vertices emerging from a vertex or from an edge the following lower bounds are easily obtained: 8 g P >< i=0 ν(k; g) ν l (k; g) = g 1 P >: 1+ (k 1) i = (k 1)(g=) k if g is even, i=1 k(k 1) i 1 = k(k 1)(g 1)= k if g is odd. Improving on previous results of Sauer [8], Lazebnik, Ustimenko and Woldar [5] recently obtained the following upper bounds (1) ν(k; g)» kq 3g 4 a ; () where q is the smallest odd prime power satisfying k» q and a = 4; 11 ; 7 ; 13 for 4 4 g 0; 1; ; 3 (mod 4) respectively. Fu, uang and Rodger [3] established that ν(k; ) is a monotone function of the girth. More precisely, they showed that For 3» k» g 1» g wehave ν(k; g 1 ) <ν(k; g ): (3) In the special case when ν(k; g) = ν l (k; g), the (k; g)-cages are called Moore graphs when g is odd and generalized polygons if g is even. It is known that Moore graphs exist only for k = (cycles), g = 3 (complete graphs) or g =5andk =3; 7 or (possibly) 57; see [4]. On the other hand, for even girth, generalized polygons with g = 4 are the complete bipartite graphs; when k 1 is a prime power, known examples of (k; g) cages are the incidence graphs of projective planes for g =6,of generalized quadrangles for g = 8 and of generalized hexagons for g =1. In this paper we prove the following result. Theorem 1 Let g f6; 8; 1g and k 3. Let q be the smallest prime power greater than or equal to k. Then ν(k; g)» kq g 4 : Moreover, 8 < ν(k; g)» : k(k 1) g 4 ( 7 6 ) g 4 if 375 k 7, k(k 1) g 4 (1 + 1 ln (k) ) g 4 if k 376.

3 UPPER BOUNDS AND CONNECTIVITY OF CAGES 3 Theorem 1, combined with the monotonocity ofν(k; ), improves the upper bound () given by Lazebnik, Ustimenko andwoldar when g = 11 and g =1. It is worth mentioning that, by using similar arguments as in [1], the construction in the proof of Theorem 1 for graphs of girth g = 6 can be extended to degrees k close to q still keeping their girth exactly six. Concerning the structure of cages, an interesting problem is that of their vertexconnectivity. Fu, uang and Rodger [3] conjectured that every (k; g)-cage is k- connected and they proved the statement for k = 3. Marcote, Balbuena and Pelayo [6] show thatevery (k; g)-cage is k-connected when g =6org = 8 and also showed that some (k; 5)-cages have connectivity k, including the cases when k 1 is a prime power. To prove their results these authors prove that a connected graph G with minimum degree ffi 3, girth g and order n is k-connected for» k» ffi if n» ν l (k; g) k; see Theorem 1 of [6]. The next proposition is a direct consequence of this result, and for the convenience of the reader we include a short proof. Proposition Let G be a graph with minimum degree ffi(g) =k 3, girth g(g) =g and vertex-connectivity»(g)» k 1. Then jv (G)j ν l (k; g)»(g). We also prove the following result. Theorem 3 Let G be a(k; g)-cage. If i) g =1,or ii) g f7; 11g and k 1 is a prime power, then G is k-connected. Therefore we confirm the conjecture of Fu, uang and Rodger for the value g = 1 and for infinitely many values of k when g = 7 or 11. Notation In what follows G denotes an undirected graph with no loops and no multiple edges. For each vertex x V (G), N G (x) and d G (x) denote the set of neighbors and the degree of x in G respectively. The minimum degree of G will be denoted as ffi(g). Givenapairofvertices x; y V (G), by an(xy)-path we mean a sequence x 0 ;:::;x r of vertices of G such thatx 0 = x, x r = y and, for every 0» i» r 1, x i x i+1 E(G). If x = y, an (xy)-path is a cycle. The length of an (xy)-path (or cycle) x 0 ;:::;x r is r. Given two vertices x; y V (G) thedistance between x and y is the minimum length of an (xy)-path in G, and will be denoted by D G (x; y). A matching is a 1- regular graph. For a subset S V (G) we denote by G[S] the subgraph of G induced by S. A subset S ρ V (G) will be said to be independent if no two vertices in S are adjacent

4 4 GABRIELA ARAUJO ET AL. in G. Given x V (G), an (x; S)-path is an (xy)-path where y S and we denote D G (x; S) = minfd G (x; y) :y Sg. For simplicity, ife = wy is an edge of G and x V (G), we will write (e; x)-path and D G (e; x) instead of (fw; yg;x)-path and D G (fw; yg;x), respectively. Let D be a directed graph. The out-degree of x is the cardinality of the set fy V (D) :(x; y) A(D)g. An oriented cyclec of D is a sequence x 0 ;x 1 ;:::;x r 1;x r = x 0 of vertices such that, for each 0» i» r 1, (x i ;x i+1 ) A(D). We shall use the following result on the existence of prime powers in short intervals of integers; see e.g. Dusart []. If k 375 then the interval [k; k(1 + 1 )] contains a prime number: (4) ln (k) If 6» k» 376 then the interval [k; 7k ]contains a prime power: (5) 6 3 Proofs of the Theorems Proof of Theorem 1. Let k 3. If k 1isaprimepower, ν l (k; g) =ν(k; g)andthe result follows. Let us suppose that k 1 is not a prime power. Let q be the smallest prime power greater than or equal to k and let g f6; 8; 1g. Let G =(V; E) bea (q +1;g)-cage, that is, a (q +1;g)-graph of minimum order. For a given edge e = xy of G and for each i 0, let N i (e) =fz V (G) :D G (e; z) =ig: The subgraph spanned by thevertices within distance l» g from fx; yg isatree as shown in Figure 1. Therefore, for 0» i» g,wehave jn i (e)j =q i : Since q is a prime power and G is a (q +1;g)-cage, V (G) = [ N i (e): 0»i» g This implies that the subgraph of G induced by N g (e) isaq-regular graph (see an illustration in Figure 1.) Let N G (x) n y = fx 1 ;:::;x q g and N G (y) n x = fy 1 ;:::;y q g; and, for each 1» i» q; let B(x i )=fz N g (e) :D G (x i ;z)= g 4 g and B(y i )=fz N g (e) :D G (y i ;z)= g 4 g:

5 UPPER BOUNDS AND CONNECTIVITY OF CAGES 5 N 3 (e) N (e) N 1 (e) N 0 (e) ffi fflr r r r r r r r r r r r r r r r B r r fl fi ffl ffi fi B B r r r r r r B r B B B B B fl ffl ffi fi B r B B B B B fl ffl ffi fi B(x ) B(x ) B(y ) B(y ) 1 1 r B B B B fl r J JΩ ΩΩ J JΩ r ΩΩ J JΩ ΩΩ J x JΩ r ΩΩ 1 x y 1 y ZZ ZZ Z Zρ ρρρ Z Zρ ρρρ x e y Figure 1: Tree of neighbors emerging from an edge for k = 3 and g =8. For each vertex z N g (e) there is exactly one (e; z)-path of length g in G. It follows that fb(x 1 );:::;B(x q );B(y 1 );:::;B(y q )g is a partition of N g (e). In particular, jb(x i )j = jb(y i )j = q g 4 ; i =1;:::;q: Note that the set U = B(x 1 ) [ :::[B(x q ) is an independent set, and similarly W = B(y 1 ) [ :::[B(y q ) is also an independent set. Furthermore, for 1» i; j» q, a vertex in B(x i ) can be joined by an edge to at most one vertex in B(y j ), since otherwise the graph would contain a cycle of length at most g 1. Since the subgraph of G induced by U [W = N g (e) isq-regular, the subgraph G[B(x i );B(y j )] induced by B(x i ) [ B(y j )isamatching for each i; j with 1» i; j» q. For each 1» k» q let G k be the subgraph induced by S k i=1(b(x i ) [B(y i )). By the above remarks, G k is k-regular, has order kq (g 4)= and, being a subgraph of G, it has girth g k g. In particular we have ν(k; g k )» kq (g 4)= for each k =3;:::;q. Since g» g k, by (3)we have that ν(k; g)» kq (g 4)= : (6) Finally, since k 1isnotaprimepower, by (4) and (5) we see that, for k 1 375, there is a prime in the interval [k; (k 1)(1+ 1 )] and, for 375 k 1 6, ln (k) there is a prime power in the interval [k; (k 1)7 ]. Therefore, 6 q» ( (k 1)7 if 7» k» 376; 6 (k 1)(1 + 1 ) if k 376: ln (k) Substitution of the above inequalities in (6) gives the second inequality in Theorem 1. Proof of Proposition. Let G be a graph with girth g(g) =g, minimum degree ffi(g) =k 3 and connectivity»(g)» k 1.

6 6 GABRIELA ARAUJO ET AL. For each vertex x V (G) define B(x) =fz V (G) :D G (x; z)»b g 1 cg; and, for each y N G (x), let B x (y) be the set of vertices z of G such that there is a (zy)-path in G of length at most b g 1 c 1 which does not use the edge xy. Since g(g) =g, the set fb x (y) :y N G (x)g is a partition of B(x) n x for each x V (G) (see an ilustration in Figure ). ' B r r x (y r 1 ) r$ ' B r r x (y r ) r$ ' B r r x (y r 3 ) r$ CC CC CC CC CC CC C CΛr ΛΛΛ AA A A AA AA y r Ar A A 1 y y 3 A & r % & %& Φ % Φ rφ ΦΦΦΦ x Figure : Tree emerging from a vertex x for b g 1 c = 3 and k =3. Let S V (G) beaminimum cutset of G and let be a connected component of G n S. Since ffi(g) = k >»(G) = jsj, each vertex x V (G) has a neighbor y N G (x) such that B x (y) S = ;. In particular, for each x V (), there is y N G (x) V () such that B x (y) V (). Let D be a digraph defined as follows: the vertex set of D is V (D) = V () and (x; y) is an arc of D if and only if B x (y) V (). By the above remark, the minimum out-degree of D is at least 1. It follows that D contains an oriented cycle C =(x 1 ;:::;x t ), t. We claim that minfd G (S; x i ); i =1;:::;tg b g 1 c: (7) Suppose on the contrary that m = minfd G (S; x i ); i =1;:::;tg»b g 1 c 1; say D G (S; x i+1 )=m. Let P bea(s; x i+1 )-path in G of length m. Since (x i ;x i+1 )isan arc of D, wehave B xi (x i+1 ) V (). Therefore P must use the edge x i x i+1, which implies D G (x i ;S)=m 1, a contradiction. This proves (7). By (7), each connected component of G n S contains at least two adjacent vertices x; y such that minfd G (S; x);d G (S; y)g b g 1 c:

7 UPPER BOUNDS AND CONNECTIVITY OF CAGES 7 It follows that B(x) [B(y) V () [ S. Suppose that g is even. Then B(x) [B(y) =fz V (G) :D G (xy; z)»b g 1 cg. By comparing with (1) we have jb(x) [B(y)j ν l (k; g). This implies that each connected component ofgns has order at least ν l (k; g) jsj. Suppose now thatg is odd. Then it is clear that jv ()j jb(x)j jsj and again jb(x)j ν l (k; g), which implies that each connected component ofg n S has order at least ν l (k; g) jsj. In both cases we get jv (G)j ν l (k; g) jsj: Proof of Theorem 3. Let G be a (k; g)-cage. The statement will follow from Proposition if we show that jv (G)j < ν l (k; g) (k 1) whenever g = 1, or whenever g f7; 11g and k 1 is a prime power. i) Let g = 1, and let q be the smallest prime power greater than or equal to k 1. If q = k 1 then G is a minimal (k;1)-cage and jv (G)j = ν l (k; 1) < ν l (k;1) (k 1). Suppose that k 1isnotaprimepower. Since ln (k)» 7 6 whenever k 376, by Theorem 1 we may assume that jv (G)j»k(k 1) 4 ( 7 6 )4 and so jv (G)j < 4k(k 1) 4. Using (1) we have jv (G)j < 4k(k 1) 4 =4((k 1) 5 +(k 1) 4 )» 4((k 1) 5 +(k 1) 4 +(k 1) 3 ) (k 1)» ν l (k;1) (k 1): ii) Let g f7; 11g and suppose that k 1 is a prime power. By (3) we have jv (G)j <ν(k; g + 1). Since k 1isaprimepower we have On the other hand and since g+1 (k 1) jv (G)j <ν l (k; g +1)= : k ν l (k; g) (k 1) = g 1 k(k 1) 4 (k 1) k = g+1 (k 1) +(k 1) g 1 4 (k 1) k = g 1 (k 1) ν l (k; g +1)+ (k 1); k (k 1) g 1 k g 3 (k )(k 1) +(k 1) g 3 = k g 3 = (k 1) g 3 (k 1) + k > (k 1)

8 8 GABRIELA ARAUJO ET AL. it follows that ν l (k; g +1) < ν l (k; g) (k 1) and so jv (G)j < ν l (k; g) (k 1). References [1] G. Araujo, M. Noy and O. Serra, A geometric construction of large vertex transitive graphs of diameter, J. Combin. Math. Combin. Comput. 57 (006), [] P. Dusart, The k th prime is greater than k(ln k +lnlnk 1) for k, Math. Comp. 68 no. 5, (1999), [3]. Fu, K. uang and C.A. Rodger, Connectivity of Cages, J. Graph Theory 4 (1997), [4] A.J. offman and R.R. Singleton, On Moore graphs with diameters and 3, IBM J. Res. Dev. 4 (1960), [5] F. Lazebnik, V.A. Ustimenko and A.J. Woldar, New upper bounds on the order of cages, Elec. J. Combin. 14 R13, (1997), [6] X. Marcote, C. Balbuena and I. Pelayo, On the connectivity of cages with girth five, six and eight, Discrete Math. Krakov 00, in press. [7] G. Royle, Cages of higher valency, [8] N. Sauer, Extremaleigenschaften regulärer Graphen gegebener Taillenweite, I and II, Sitzungsberichte Österreich. Acad. Wiss. Math. Natur. Kl., S-B II, 176 (1967), 9 5; 176 (1967), [9] P.K. Wong, Cages A Survey, J. Graph Theory 6 (198), 1. (Received 17 May 006; revised 5 Nov 006)