On the Connectivity of (k, g)-Cages of even

Yuqing Lin School of Electrical Engineering and Computer Science The University of Newcastle, NSW2308, Australia e-mail: [email protected]

Camino Balbuena, Xavier Marcote Department de Matem`aticaAplicada III Universitat Polit`ecnicade Catalunya e-mail: {m.camino.balbuena,francisco.javier.marcote}@upc.edu

Mirka Miller School of Information Technology and Mathematical Sciences University of Ballarat, Ballarat, Victoria 3353, Australia e-mail: [email protected]

Abstract

A(k,g)-cage is a k- with girth g and with the least possible number of vertices. In this paper we give a brief overview of the current results on the connectivity of (k,g)-cages and we improve the current known best lower bound on the vertex connectivity of (k,g)- for g even.

1 Definitions

Throughout this paper, only undirected simple graphs without loops or mul- tiple edges are considered. Unless otherwise stated, we follow [4] for termi- nology and definitions. The vertex set (respectively, edge set) of a graph G is denoted by V (G) (respectively, E(G)). If V 0 is a nonempty subset of V then the subgraph induced by V 0 is denoted by G[V 0]. Similarly, if E0 is a nonempty subset

1 of E then the subgraph induced by E0 is denoted by G[E0]. The subgraph obtained from G by deleting the vertices in V 0 together with their incident edges is denoted by G − V 0. The graph obtained from G by deleting a set of edges E0 is denoted by G − E0. The set of vertices adjacent to a vertex v is denoted by N(v). If X is a S nonempty subset of vertices, then N(X) stands for the set ( x∈X N(x))\X. The degree of a vertex v is deg(v) = |N(v)|, and a graph is called regular when all the vertices have the same degree. The minimum degree of the graph is denoted by δ. The distance d(u, v) between two vertices u and v in V (G) is the length of the shortest path between u and v. We also use the notion of a distance between a vertex v and a set of vertices X, written d(v, X); it is the distance from v to a closest vertex in X. The length of a shortest cycle in a graph G is called the girth of G.A k- regular graph with girth g is called a (k, g)-graph.A(k, g)-graph is called a (k, g)-cage if it has the least possible number of vertices. Throughout this paper, f(k, g) stands for the order of a (k, g)-cage. A graph G is connected if there is a path between any two vertices of G. Suppose that G is a connected graph. We say that G is t-connected if the deletion of at least t vertices of G is required to disconnect the graph. Sim- ilarly, we say that a graph is t-edge-connected if the deletion of at least t edges of G is required to disconnect the graph. A vertex-cut (respectively, edge-cut) of a graph is a set of vertices (respectively, edges) whose removal disconnects the graph. A graph is maximally connected (respectively, max- imally edge-connected) if the minimum cardinality of a vertex-cut (respec- tively, edge-cut) is equal to the minimum degree of the graph. An edge-cut W is called trivial if it contains all the edges incident with some vertex, that is, {xxi ∈ E(G): xi ∈ N(x)} ⊆ W for some x ∈ V (G). A maximally edge-connected graph is edge-superconnected if all its minimum edge-cuts (with cardinality equal to δ) are trivial. A vertex-superconnected graph is defined similarly.

2 Overviews

Cages were introduced by Tutte [29]. For example, the cycle Cg is the unique (2, g)-cage, the Kn is the unique (n − 1, 3)-cage, the complete Kn,n is the (n, 4)-cage, the is the

2 (3, 5)-cage, the Hoffman-Singleton graph is the (7, 5)-cage, and the is the (3, 6)-cage. Note that cages are not necessarily unique for each given pair of values k and g. For instance, there exist 18 non-isomorphic (3, 9)-cages, all of order 58 [3]. In 1963, Erd¨osand Sachs [7] proved that (k, g)-cages exist for any given value of the pair (k, g). Since cages are specified as a set of desired properties rather than by concrete detailed construction rules, determining the graph for given value of k and g becomes an interesting research topic. Moreover, this topic is very challenging; even to estimate the bounds on the order of a (k, g)-cage is very difficult in general when g ≥ 5 and k ≥ 3. It is obvious that a of given degree ∆ and diameter D is a (∆, 2D + 1)-cage for odd girth g = 2D + 1, and a is a (∆, 2D)-cage for even girth g = 2D. Thus existence or non-existence results related to the Moore bound and generalized polygons give the lower bounds on the order of (k, g)-cages. By inspecting a k-regular tree with diameter b(g − 1)/2c, it is easy to establish the following lower bounds for the order of a (k, g)-cage G:

 (g−1)/2  k(k−1) −2 (g odd); |V (G)| ≥ k−2  2(k−1)g/2−2 k−2 (g even).

A(k, g)-cage which reaches this lower bound is also called a minimal (k, g)- graph. Due to the results of Hoffman and Singleton [14], Damerell [5] and Bannai and Ito [2] on the existence of Moore graphs, we know that minimal (k, g)-graphs for odd girth exist when:

• g is equal to 5 and k is equal to 3, 5, or possibly 57.

Furthermore, the results of Feit and Higman [8] on generalized polygons prove that minimal (k, g)-graphs for even girth exist only when:

• g is equal to 6, 8 or 12.

In [7], Erd¨osand Sachs also established the first upper bound for f(k, g). The results were improved by Sauer [26] as follows:   29 g−2 4  ( 12 )2 + 3 (g odd); f(3, g) ≤  29 g−2 2 ( 12 )2 + 3 (g even).

3  g−2  2(k − 1) (g odd); For k ≥ 4 : f(k, g) ≤   4(k − 1)g−3 (g even).

The upper bound obtained by Erd¨osand Sachs is roughly equal to the square of the lower bound. Currently the best-known upper bounds are roughly the lower bounds to the power of 3/2 and, in [18], the authors provide an explicit construction for such (k, g)-graphs. As we can see, there is still a large gap between the lower bound and the upper bound. Large amount of work has been carried out on sharpening the lower and upper bounds, especially for small values of k and g. The current status of the problem can be obtained from the website maintained by Royle [25]. So far, very little is known about general structural properties of cages. The first fundamental property of cages, known as the ‘girth monotonicity theorem’, was established by Erd¨osand Sachs in [7], and independently by Holton and Sheehan [15], Fu, Huang and Rodger [9].

Girth Monotonicity Theorem [7, 9, 15] If k ≥ 3 and 3 ≤ g1 < g2 then f(k, g1) < f(k, g2). This theorem has been used extensively in investigating the connectivity of cages. Recently, Wang and Yu [32] proved the following result regarding degree monotonicity:

Theorem 1 [32] Let f(k, g) be the order of a (k, g)-cage. Then f(k, g) ≤ f(k + 1, g) for g odd.

Theorem 2 [32] Let f(k, g) be the order of a (k, g)-cage. Then f(k, g) ≤ f(k + 2, g).

Clearly, any results concerning degree monotonicity, especially if it involves strict inequality, would be a great help in the task of investigating cages. There are results concerning other structural properties of (k, g)-cages, for example, on the diameter of (k, g)-cages [7], on the size of the cycle and the number of cycles containing a particular edge [16]. There are also many conjectures on the structure of cages, for example, Harary and Kov´acs[13] conjectured the existence of a (k, g)-cage containing cycles of length g+1 for each odd girth g ≥ 5 and each k ≥ 3. There is also a conjecture stating that

4 all cages of even girth are bipartite [31]. However, settling these conjectures seems to be a very challenging task. The study of the connectivity of cages has been suggested by several au- thors. Especially, in [9], Fu, Huang, and Rodger have posed the following conjecture.

Conjecture 1 [9] Every (k, g)-cage is maximally connected.

This conjecture also implies that (k, g)-cages are maximally edge connected. In the following sections, we will summarize the known results related to this conjecture and also provide some new results.

2.1 Edge connectivity of (k, g)-cages

Concerning the edge-connectivity of cages, which is viewed as a softer version of Conjecture 1, many results are available. In [30], Xu et al. proved the following theorem:

Theorem 3 [30] All (k, g)-cages of odd girth g ≥ 5 are k-edge-connected.

Later, Lin et al. citeYMC proved that:

Theorem 4 [19] All (k, g)-cages of even girth g ≥ 6 and degree k ≥ 3 are k-edge-connected.

The same results have also been obtained independently by other groups of researchers, including Balbuena and Marcote [22]. Combining the above two theorems, we conclude that (k, g)-cages are k-edge-connected. In other words, (k, g)-cages are maximally edge-connected, and the problem is com- pletely solved. Because of their interest in the structure of edge-cuts, Balbuena and Mar- cote suggested the study of the edge superconnectivity. The intention is to prove that all minimal edge-cuts are trivial. In [22], they proved that, when g is even, (k, g)-cages are edge-superconnected. Subsequently, Lin et al. [21] proved that, when g is odd, (k, g)-cages are edge-superconnected. Combining these two theorems allows us to conclude that:

5 Theorem 5 [21] All (k, g)-cages are edge-superconnected (all the minimum edge-cuts are trivial).

2.2 Connectivity of (k, g)-cages

Turning our attention to the vertex connectivity of (k, g)-cages, we find that results are far from satisfactory. The very first set of results was established around the period of 1997 and 1998. It was shown independently by Fu, Huang, and Rodger [9], by Jiang and Mubayi [17], and by Daven and Rodger [6] that:

Theorem 6 The following statements hold:

1. [9] Every (k, g)-cage is 2-connected.

2. [9] Every (3, g)-cage is 3-connected.

3. [6, 17] Every (k, g)-cage with k ≥ 3 is 3-connected.

In [30], it was shown that all (4, g)-cages are 4-connected. Recently, Marcote et al. showed that (k, g)-cages with g ≥ 10 are 4-connected [24]. Further- more, they showed that (k, g)-cages with girth g = 6, 8 are k-connected and that most cages with g = 5 are also k-connected [23]. In [20], Lin et al. proved the following theorem:

Theorem 7 [20] Let G be a (k,√ g)-cage with k ≥ 3 and odd girth g ≥ 7. Then G is r-connected with r ≥ k + 1.

Using the same technique as the one used in [20], we can also obtain some results about connectivity of cages when g is even. In our proof, we shall use the following lemma which has been used extensively in investigating the connectivity of any graph with a prescribed girth.

Lemma 1 [1, 10, 11, 27, 28] Let G = (V,E) be a graph with minimum degree k and girth g. Assume that X ⊂ V is a cutset with cardinality |X| ≤ k − 1. Then, for any component H of G − X, there exists some vertex u ∈ V (H) such that d(u, X) ≥ b(g − 1)/2c.

6 Theorem 8 Let G be a (k, g)-cage with k ≥ 3 and even girth g ≥ 6. Then G is (r + 1)-connected, r being the largest integer such that r3 + 2r2 ≤ k.

Proof. Taking into account Theorem 6, we can assume that r ≥ 3, so r2 > r + 1. We reason by contradiction. Suppose that S is a minimum vertex-cut of G of cardinality at most r, where r is the largest integer satisfying r3 +2r2 ≤ k. Let C be the smallest component of G − S. We partition the set S into the following three subsets.

X = {s ∈ S : |N(s) ∩ V (C)| ≤ r}

Y = {s ∈ S : r + 1 ≤ |N(s) ∩ V (C)| ≤ r2}

Z = {s ∈ S : |N(s) ∩ V (C)| ≥ r2 + 1}

It is easy to see the following facts:

• |N(X) ∩ V (C)| ≤ r2.

• |N(Y ) ∩ V (C)| ≤ r3.

• |Y ∪ Z| ≤ r.

• |(N(X) ∩ V (C)) ∪ Y ∪ Z| ≤ r2.

Obviously, the set (N(X) ∩ V (C)) ∪ Y ∪ Z is also a vertex-cut whose car- dinality is smaller than k. Instead of considering the vertex-cut S, we shall consider this new vertex-cut (N(X) ∩ V (C)) ∪ Y ∪ Z. By Lemma 1, we know that there exists a vertex u in C such that the distance from u to (N(X) ∩ V (C)) ∪ Y ∪ Z is at least g/2 − 1. From now on, we shall use the term “short distance” to mean that the distance between two vertices is less than g/2 − 1. Moreover, let W stand for the set of edges of the subgraph of G induced by Y ∪ Z, that is, W = E(G[Y ∪ Z]). It is easy to see that there are at most r3 vertices in N(u) which have distance g/2 − 1 or g/2 − 2 to Y in the graph G − W . This is because all the

7 shortest paths in G − W of length g/2 − 1 or g/2 − 2 from vertices in N(u) to vertices in Y must go via the vertices in N(Y ) ∩ V (C). We know that |N(Y ) ∩ V (C)| ≤ r3, therefore, there are at most r3 vertex disjoint paths of length g/2 − 1 or g/2 − 2 from N(u) to Y . Otherwise, by the Pigeonhole Principle, there exists a cycle of length less than g in the graph (going through u, two distinct vertices in N(u), and some vertex in N(Y ) ∩ V (C)), a contradiction. It is also easy to see that |(N(X) ∩ V (C)) ∪ Z| ≤ r2, because |(N(X) ∩ V (C)) ∪ Z| ≤ |X|r + |Z| and |X ∪ Z| ≤ r. Applying Pigeonhole Principle again as in the previous case, we know that, among the leftover at least k−r3 vertices in N(u), there are at most r2 vertices which have distance g/2 − 2 in G to (N(X) ∩ V (C)) ∪ Z. Therefore, there are at least k − r3 − r2 ≥ r2 vertices in N(u) which have distance at least g/2 to Y and at least g/2 − 1 to (N(X) ∩ V (C)) ∪ Z in G − W . Denote arbitrary r2 of these vertices by U = {u1, u2, . . . , ur2 }. See Fig. 1.

N (X) C C S X

degree ≤ r

Y

u1 u

r + 1 ≤ degree ≤ r2

Z

u 2 U r degree ≥ r2 + 1

Figure 1: Structure of component C and S

8 For each vertex ui in U, denote by Ui the vertices in N(ui) − u which have distance at least g/2−1 to (N(X)∩V (C))∪Y ∪Z in G−W . It is clear that 2 2 the cardinality of Ui is at least k−r −1, since |(N(X)∩V (C))∪Y ∪Z| ≤ r . c Denote by Ui the set of vertices in N(ui) − u which have distance at least c c g/2 − 1 to Y ∪ Z in G − W , so Ui ⊆ Ui. It is easy to see that |Ui| ≥ k − r − 1 since |Y ∪ Z| ≤ r. We shall take the subgraph of G − W induced by V (C) ∪ Y ∪ Z − {u} and delete some vertices, denoting the resulting graph by H. We shall take another copy of H and denote it by H0, the corresponding sets being 0 0 0 0 0 0 0 U = {u1, u2, . . . , ur2 }, Y and Z , denoting by C the copy of C. We will join the vertices of H and H0 to construct a new graph G0. The new graph will be regular of degree k and girth at least g but with fewer vertices than G. By the Girth Monotonicity Theorem, we know that the resulting graph should have no fewer vertices than G, arriving at a contradiction which proves the theorem. The connections are described below, see Fig. 2 for illustration. 1) We do not know the degree of each vertex in N(X) ∩ V (C) at this point. However, we know that the number of new edges that should be added in order to achieve degree k for all the vertices in N(X) ∩ V (C) is at most |X|r ≤ r2 − |Y ∪ Z|r ≤ r2 − |Y ∪ Z| ≤ |U 0|. Therefore, every vertex, say 0 xi, in N(X) ∩ V (C) will be matched to k − |N(xi) ∩ V (C)| vertices in U and connected to all of them. We shall make the same connections between N(X0) ∩ V (C0) and U. It is obvious that now the vertices in N(X) ∩ V (C) and N(X0) ∩ V (C0) have degree k. 2) There are at least |Y |+|Z| vertices left in U which have degree k−1. Every 0 0 vertex yi in Y will be arbitrarily matched with a vertex in U , say ui. We will 0 c0 remove ui from the graph and connect yi to some of the vertices in Ui such c0 that yi has degree k. Note, |Ui | ≥ k−r−1 and k−|N(yi)∩V (C)| ≤ k−r−1. Therefore, we can guarantee that the degree of yi equals to k by connecting c0 0 it to vertices in Ui . We shall make the same connections between Y and U. 3) At this stage, there are at least |Z| vertices left over in U with degree k − 1. Every vertex zj in Z will be arbitrarily matched with a vertex in 0 0 0 U , say uj. We will remove uj from the graph and connect zj to some of 0 0 2 the vertices in Uj such that zj has degree k. Also note, |Uj| ≥ k − r − 1 2 and k − |N(zj) ∩ V (C)| ≤ k − r − 1. Therefore, we can guarantee that the 0 degree of zj is equal to k by connecting it to vertices in Uj. We shall make the same connections between Z0 and U.

9 4) All the rest of the vertices in the graph have degree k or k−1. We connect

each vertex x ∈ V (H) having degree k − 1 to its copy x0 ∈ V (H0).

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0 ¨ © 0

N (X) ¢ N (X )

C £ C

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 U U 0











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Y Y 0

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Z Z0

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Figure 2: Illustration of the construction of G0

Now it is easy to see the new graph G0 is regular with degree k. We verify next that this graph has girth at least g. It is clear that any cycle completely contained in either H or H0 has length at least g, and that any other cycle in G0 must contain m = 2q ≥ 2 new edges. Let us first consider a cycle C in G0 containing exactly two new edges, say ab0 and cd0, with a, b ∈ V (H), b0, d0 ∈ V (H0)(x0 stands for the copy in V (H0) of vertex x, for every x ∈ V (H)). Note that C also contains a path in H connecting vertices a and c, and another path (in H0) connecting b0 and d0. We denote the length of C by `, and consider the following cases: Case 1: If C goes through the two added edges ac0 and ad0 (that is, b = a), then the length of this cycle is at most 2 + (g − 2) = g, because c0 and d0 were taken to be neighbors of some vertex which has been finally deleted. Hence, we can consider pairs of independent new edges from now on. Case 2: Suppose that the two considered edges are aa0 and bb0 (c0 = a0, d0 = b0). In this case, each of the vertices a, b must belong either to N(u) or

10 0 0 to N(ui) − ui, for some ui ∈ U that has been deleted (similarly for a , b ). Then ` ≥ 2 + 2(g − 4) ≥ g. Case 3: The two considered edges are aa0 and bd0, with d0 6= b0. First, if a ∈ N(u) then the cycle has length ` ≥ 2 + (g − 3) + (g/2 − 2) ≥ g. Second, if a ∈ N(ui) − ui, for some ui ∈ U, then the length ` of the cycle C is not less than: 2 + (g/2 − 2) + (g − 3) ≥ g, if b ∈ N(X) ∩ V (C) or d0 ∈ N(X0) ∩ V (C0); 2 + (g/2 − 1) + (g − 4) ≥ g, otherwise.

b Case 4: For arbitrary vertex ui in U, the vertices in Ui have distance g/2−1 0 0 to Z; similarly, the vertices in Uj will have distance g/2−1 to Y . Therefore, b 0 0 0 0 if a ∈ Ui, b ∈ Z, c ∈ Y and d ∈ Uj, then ` ≥ 2 + 2(g/2 − 1) = g. b 0 b 0 0 0 Case 5: Let a ∈ Y , b ∈ Ui, c ∈ Uj and d ∈ Y . Then ` ≥ 2+2(g/2−1) = g. 0 0 0 0 Analogously, when a ∈ Z, b ∈ Ui, c ∈ Uj and d ∈ Z , we have ` ≥ 2 + 2(g/2 − 1) = g. 0 b 0 0 0 b 0 0 Case 6: Let a, b ∈ Y ∪ Z, so c ∈ Ui ∪ Uj and d ∈ Uα ∪ Uβ. Hence, ` ≥ 2 + 2 + (g − 4) = g.

0 0 c0 Case 7: Consider that C goes through the edge ac with a ∈ Y and c ∈ Ui , and through the edge bd0 with b ∈ U and d0 ∈ N(X0) ∩ V (C0). The vertices c0 0 c0 0 in Y are connected to some of the vertices in Ui −Ui , and vertices in Ui −Ui have distance g/2 − 1 to Y 0 ∪ Z0, but maybe distance g/2 − 2 to vertices in N(X0) ∩ V (C0); however, the vertices in N(X0) ∩ V (C0) are connected to vertices in U, which have distance at least g/2 to Y . Therefore, the cycle will have length ` ≥ 2 + (g/2 − 2) + g/2 = g. Case 8: It is obvious that any cycle going through one edge in between the 0 c0 0 set N(X) ∩ V (C) and U and another edge in between Y ∪ Z and Ui ∪ Uj has length not less than 2 + (g − 3) + 1 = g. Case 9: Let a ∈ N(X) ∩ V (C), c0 ∈ U 0. Suppose first b ∈ U, then d0 ∈ (N(X0) ∩ V (C0)) ∪ U 0, so ` ≥ 2 + (g/2 − 1) + (g/2 − 1) = g. Second, if we assume that b ∈ N(X) ∩ V (C), then C has length ` ≥ 2 + (g − 2) + 1 > g.

0 0 0 0 0 Case 10: Suppose a ∈ Z, b ∈ U, c ∈ Uj and d ∈ N(X ) ∩ V (C ). In this case ` ≥ 2 + (g/2 − 1) + (g/2 − 1) = g. So every cycle in G0 containing exactly two new edges has length at least g. Let us next consider a cycle C in G0 containing m = 2q ≥ 4 new edges, and

11 let us call A ⊂ V (H) and B0 ⊂ V (H0) the sets of end vertices of these m new edges. Clearly, the cycle C must also contain q ≥ 2 vertex-disjoint paths in H connecting vertices in A, and q ≥ 2 vertex-disjoint paths in H0 connecting vertices in B0. Observe that each vertex in A ∪ B0 can be incident with at most two of the m new edges of C. If ab0, ad0 are two of these new edges, then b0, d0 ∈/ (N(X0)∩V (C0))∪Y 0 ∪Z0, hence b0, d0 are only incident with the new edges ab0, ad0 respectively. This means that C contains two disjoint paths in H0 of length at least g/2−2, then the length of the cycle is ` ≥ 4+2(g/2−2) = g. Thus, we can assume for the rest of the proof that all vertices in A ∪ B0 are incident with only one of the m new edges of C; that is, |A| = |B0| = m. If any of the paths in H or H0 contained in C connects two distinct vertices S 0 0 S 0 0 in either F = N(u) (N(ui) − ui) or F = N(u ) 0 0 (N(u ) − u ), ui∈U ui∈U i i then the length of the cycle is ` ≥ 4 + (g − 4) = g. Next, we continue by assuming that |F ∩ A| ≤ |A|/2 = q and |F 0 ∩ B0| ≤ |B0|/2 = q. If it were |A \ F | > q, then |F 0 ∩ B0| ≥ |A \ F | > q (because there is an injective map from A \ F to F 0 ∩ B0, defined by the considered new edges in C), contradicting our last assumption |F 0 ∩ B0| ≤ q. As a consequence, |A \ F | ≤ q, hence |A \ F | = |F ∩ A| = q = |A|/2. Then, there exist q = m/2 paths in C ∩ H connecting one vertex in F with some other vertex in A \ F , so all of them have length at least g/2 − 2 ≥ 1. This means that the length of C is at least m + (m/2)(g/2 − 2) = mg/4 ≥ g. Therefore, the length of C is ` ≥ m + (m/2)(g/2 − 2) = mg/4 ≥ g. Therefore, we have constructed a (k, g0)-graph G0, having girth g0 ≥ g and less vertices than the graph G, arriving at a contradiction.

Remark: In this paper we have chosen to present our constructions and arguments in the simplest forms. It is clear that in several places we could improve the above results by some constant. It is a matter of carefully examining the cardinality of Y and X, but at the expense of unwelcome obscurity and excessive length.

3 Open problems

Some of the interesting directions and open problems have already been highlighted in this paper. Currently, Conjecture 1 is still open. Regarding edge connectivity, naturally, one might ask the following question:

12 Open Problem 1 What is the minimum size of a non-trivial edge-cut?

One might also investigate more refined connectivity or edge connectivity related measurements on cages, such as the restricted edge-connectivity or other parameters which can be understood on the basis of the conditional connectivity introduced by Harary in [12]. Additionally, there is little known about the general structure of cages; properties such as radius and eccen- tricity of cages have not been investigated fully. It would be interesting to see some results on these topics.

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