Generalized Cages *

Let 2 k 1 < k 2 < · · · < k t , 3 g 1 < g 2 < · · · < g s < N be integer parameters. and extend the concept of a cage (a smallest graph of given degree and girth) to that of a generalized cage – a smallest (k 1 introduce several infinite families of generalized cages and study their basic properties in the context of connected, bipartite, and vertex-transitive graphs, as well as combinatorial configurations (in the context of multilaterals). * Dedicated to Branko Grünbaum whose work is a constant source of inspiration for us.


Introduction
The main motivation for our paper comes from [4] that addressed among other topics the question of the existence of trivalent graphs with certain prescribed and prohibited cycle lengths.This called, for example, for the construction of a trivalent graph of girth 6 that contains an 8-cycle, no 10-cycle and a 12-cycle, and all the possible combinations of prescribed or prohibited 6-, 8-, and 10-cycles.The authors have been able to construct a trivalent graph for any combination of these, and became interested in characterizing all possible combinations of cycle lengths for regular graphs.In the original version of this paper, we were able to show that essentially any sequence of cycle lengths can be the beginning of the cycle spectrum of a finite graph of degree k.Only after solving our original problem, we found out that a more general version of the above problem has been considered by Sachs in 1963 [22].He was able to prove the following: Theorem 1 ( [22]).Corresponding to any two integers k 3 and N 4 and any N − 2 integers α 3 , . . ., α N it is possible to construct an unlimited number of mutually nonisomorphic connected regular graphs of degree k without bridges or cut-vertices, which contain exactly α i cycles of length i for i = 3, . . ., N and in which all the cycles of length N are mutually disjoint.
In our approach, we think of the problem of finding finite graphs with prescribed degrees and girths as a generalization of the well-known (k, g)-Cage Problem [8]: Problem 2. Given integers k 2 and g 3, construct a k-regular graph of girth g of the smallest possible order.
The smallest k-regular graphs of girth g are called (k, g)-cages.The aim of our paper is to generalize the well-known problem of finding (k, g)-cages to that of finding generalized cages -smallest graphs for which we prescribe and limit both the vertex degrees and cycle lengths.Special cases of generalized cages have already been considered under several different settings.One of these directions allowed for extending the degrees considered and involved the study of graphs of prescribed bi-degree and girth [1,10] or prescribed set of degrees and girth [6,3].In another direction, Harary and Kovács introduced the problem of the construction of regular graphs with prescribed even and odd girth [13,14], a topic of considerable interest [5,2,1,16,24].Obviously, each change of the original requirements required a separate proof of the existence of finite graphs with the characteristics under consideration.From this point of view, the graphs constructed in our proofs of Corollary 8 and Theorem 10 serve as universal examples for all of the above generalizations of (the original) concept of cages.
The precise definition of generalized cages we use in this paper proceeds as follows.A (k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )-graph, 2 , is a graph that contains vertices of degrees k 1 , k 2 , . . ., k t but no other degrees and cycles of lengths g 1 , g 2 , . . ., g s but no other cycles of length < N .A (k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )-cage is defined accordingly.We need to stress that even though we specify the degrees required of our graphs, we do not specify how many of the vertices that have the required degrees (beyond the fact that we always need at least one vertex of the particular order) must exist and we only prescribe the beginning of the cycle spectrum for our graphs (hence, specifically, we do not specify whether a cycle of length N must or must not exist).Occasionally, in the case when we only want to limit the girth of the graphs up to g s , we will leave out the upper bound N , and will talk about the (k 1 , . . ., k t ; g 1 , . . ., g s ; )-graphs that are simply graphs with the required degrees k 1 , . . ., k t (and no others) and cycles of the required lengths g 1 , . . ., g s (and no others of length g s ).Similarly, the notation (k 1 , . . ., k t ; ; N ) indicates graphs of girth at least N .
The problem of finding (k, g)-cages has been quite popular during the 1960's and 70's.After this initial period, the remaining problems were deemed hard, and not much has been happening with regard to cages until the introduction of algebraic and topological methods in the 1990's.These marked a revival of the topic with the focus on the construction of infinite families of small graphs of given degree and girth (see, for example, the survey paper [8]).
Following this general trend, we focus specifically on topological constructions and as a part of our treatise we investigate a construction that is a a generalization of the truncation construction particularly popular in topological graph theory.It is also a generalization of a method introduced by Sachs in [22] which helped him to prove Theorem 1.Using this construction in combination with the voltage graph construction, we prove a positive answer to the following question for all but the 1-regular graphs: Unlike the case of the original cages (which are all connected), several generalized cages found in this paper turn out to be disconnected (as seen, for example, in the first figure of our paper).In view of this, throughout the paper we are careful to stress the difference between the results concerning connected and disconnected graphs.In the second half of our paper, in addition to proving the general existence of (k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )graphs, we obtain bounds on the order of the (k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )-cages, refine the problem to the classes of connected, bipartite, and vertex-transitive graphs, and investigate the connection of such graphs to incidence structures. 2 The cases k 1 = 1 and k 1 = 2 Before introducing our main construction, note that graphs satisfying k 1 = 1 or k 1 = 2 (graphs possessing vertices of degree 1 or 2) are very easy to understand (see, for example, the next figure).Due to their simplicity, we deal with these cases first.
(ii) A (1, 2; g 1 , g 2 , . . ., g s ; N )-graph exists for every set of parameters 3 g 1 < • • • < g s , but there are no connected (1, 2; g 1 , g 2 , . . ., g s ; N )-graphs. .., g s ; N )-graph can be deduced from the above constructions, it is ultimately not completely easy to determine.For example, a straightforward analysis of all possible cases shows that the smallest (1, 2; 3; 4)-graph is simply the 3-cycle united with a single disconnected edge, n(1, 2; 3; 4) = 5, and the smallest (1, 2, 3; 3; 4)-graph is the 3-cycle with a pendant edge attached at one of the vertices, n(1, 2, 3; 3; 4) = 4 (see Figure 1).Somewhat surprisingly, the generalized cage with a larger set of degrees and the same set of required girths happens to be of a smaller order.The case when we allow for degree 2 is of a slightly different nature.
(iii) If k 1 = 2 and t > 1, there exist connected (2, k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )-graphs for any choice of the parameters Proof.The case t = 1 follows from the fact that all 2-regular graphs consist of disjoint cycles.
If k i = 2m for m > 1, joining m cycles through a single vertex results in a vertex of degree k i and all other vertices of degree 2. If, on the other hand, k i = 2m+1, m 1, joining two bouquets of m cycles through a single edge results in two vertices of degree k i and all other vertices of degree 2. It is not hard to see that further manipulation together with choosing the right cycle lengths can result in connected (2, k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N )graphs.

Generalized truncation construction
The truncation of a map is well-known construction from topological graph theory in which one saws off the vertices of a map (an embedding of a graph into a surface) together with small immediate neighborhoods and then attaches cycles in their place [11].Our main construction is a generalization of this construction.
Let G be a finite graph with the degree set {d 1 , d 2 , . . ., d t } (i.e., the degree d v of each vertex v ∈ V (G) is equal to exactly one of the d i 's).Each edge of G can be associated with two opposing directed edges, each of which starts at other end-vertex of the edge.Let D(G) denote the set of these oriented edges; note that |D(G)| = 2|E(G)|.While orientable topological maps naturally come with an inherent ordering of the oriented edges emanating from a vertex, in the case of graphs, we are free to choose this ordering ourselves: A vertex-neighborhood labeling of G is a function ρ from the set D(G) into the set of positive integers that orders the oriented edges adjacent to the vertices of G, i.e., a function that maps the oriented edges starting from a vertex v bijectively onto the set We note that every vertex-neighborhood labeling of a connected graph G uniquely determines an orientable embedding of G [11].In that sense, the truncation of a graph G with a vertex neighborhood labeling ρ is simply a truncation of a specific embedding of G.The main difference between our truncation and the topological truncation, in which the electronic journal of combinatorics 22(1) (2015), # P1.77 vertices are always replaced by cycles, lies in the fact that we truncate by all kinds of graphs: Let H 1 , H 2 , . . ., H t be a family of graphs of orders d 1 , d 2 , . . ., d t , respectively, with the vertices of the H i 's labeled by the numbers 1, 2, . . ., |V (H i )| = d i , for 1 i t.The generalized truncation of the graph G with a vertex-neighborhood labeling ρ by the family H 1 , H 2 , . . ., H t is the graph T (G, ρ; H 1 , H 2 , . . ., H t ) obtained from G by replacing each of the vertices of G by one the graphs H i according to the following rule: Let v be a vertex of G of degree d i , and let e i , 1 i d i , be the oriented edges emanating from v labeled according to the vertex-neighborhood labeling ρ, i.e., labeled so that ρ(e i ) = i.Then, v is replaced by the graph H i of order d i by first removing the vertex v together with a part of each of the edges e i , 1 i d i , and then attaching H i to the oriented edges e i in such a way so that a vertex of H labeled by i is attached to the edge e i .An example of one such local truncation is included in Figure 3 in which a vertex of degree 4 is replaced by a labeled 4-cycle with a chord.The significance of the generalized truncation construction in constructing graphs with prescribed and prohibited degrees and cycles is summarized in the following theorem.
Moreover, if both G and the H i 's are connected, so is G, and if g min < 2g, then g min is the exact girth of G.
Proof.In G, color the new edges (the edges of the graphs H i ) red, and the old edges (the original edges of G) blue.The blue edges form a 1-factor of G, and each vertex v of G is incident with one blue edge and k j i red edges of G (for some 1 i t j , 1 j t).Thus, the degree set of G equals K = {k j i + 1 | 1 i t j , 1 j t}, as claimed.Furthermore, no cycle C of G has two consecutive blue edges, and we have two possibilities to consider.If C contains no blue edges at all, it is completely red, and its vertex set must be a subset of a copy of H i for some 1 i t.Thus, in this case, C is a cycle of H i and its length must be g i .The other possibility is that C contains both red and blue edges.For any 2-colored cycle, if we remove the red edges, the sequence of blue edges is non-repeating and any two adjacent blue edges (with a red path between them) must have been adjacent in G. Thus, the blue edges constitute a walk in the original graph G that contains at least one cycle of G, and the red/blue cycle is of length at least 2g.The last statement of our theorem follows from the fact that G contains a red cycle of length g i for each 1 i s.
The connectivity of G in the case when both G and the H i 's are connected is easy to see.
Note that the above proof shows an even stronger result.Namely, in the case when all the graphs H i are connected, each cycle C of the graph G is 'lifted' into a cycle of the graph G with the length of the lifted cycle being at least twice the length of C.
4 Disconnected (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graphs In this section, we solve the general question of the existence of (k 1 , . . ., k t ; g 1 , . . ., g s ; N )graphs.As stated in the introduction, we will construct such graph for every set of parameters 2 It is important to note that if we do not require the graph to be connected, the problem is quite a bit easier to handle.As in the case of the 2-regular graphs covered in Theorem 5, we can compose the (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graph of disconnected components, each of them being a (k; g; N )-graph, k ∈ {k 1 , . . ., k t }, g ∈ {g 1 , . . ., g s }.Our main claim thus follows from the following theorem that is a special case of the more general Theorem 1 of Sachs.Its proof provides us with a nice example of the use of the generalized truncation.
Proof.We proceed by induction on k.The case of k = 2 has been settled in Theorem 5.The graph can be taken to be a single g-cycle.
For k = 3, let G be any (g, N 2 )-graph (the existence of which is guaranteed for example by the result of [22]).Note that g is the degree of the graph while N 2 is its girth.Let H be the g-cycle, and ρ be any vertex-neighborhood labeling of G.It follows from Theorem 6 that the generalized truncation graph T (G, ρ; H) is a (3; g; N )-graph.
For the induction step, assume the existence of a (k; g; N )-graph H for some k 2 and 3 g N of order n.Let G be any n-regular graph of girth at least N 2 (guaranteed again by [22]), and let ρ be any vertex-neighborhood labeling of G. Theorem 6 asserts that the truncated graph T (G, ρ; H) is a (k + 1; g; N )-graph.
Proof.If k 1 2, the graph can be taken to be the disjoint union of graphs G i,j , where G i,j is a (k i ; g j ; N )-graph whose existence is guaranteed by Theorem 6, 1 i t, 1 j s.If t > 1, the result follows from Theorem 4 (iii).
The reader may find it a bit disappointing that we broke the intriguing problem of mutually coexisting cycles into separate problems by constructing a disconnected graph.As seen in Theorem 5, the disconnectedness of our solution may appear inherent to the problem.This is however not the case, and the only sets of parameters that do not allow for connected graphs are those from Theorems 4 and 5.In the next two sections we will construct connected (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graphs for all other parameter sets.

Voltage graph lifts
Our construction of connected (k; g 1 , g 2 , . . ., g s ; N )-graphs includes the use of a lift of a graph with a voltage assignment (see, e.g., [11]).We briefly describe this method and prove the existence of a connected graph with properties needed for our main construction.
The base graph B is allowed to have multiple edges and multiple loops and we replace each edge and loop of B by a pair of opposing (oriented) darts (with D(B) denoting the set of darts of B).A voltage assignment on B is any mapping α from D(B) into a group Γ that satisfies the condition α(e −1 ) = (α(e)) −1 for all e ∈ D(B).
The derived regular cover (lift) of B with respect to the voltage assignment α is denoted by B α , and has the vertex set V (B) × Γ (written in the form u g , u ∈ V (B), g ∈ Γ).Two vertices u g and v f are adjacent in A lift of B is said to be cyclic if the voltage group used is a cyclic group Z n of order n (which we will write in the additive notation).
We are ready to prove the main theorem of our section.
Lemma 9. Let k 2, g 3, and B be a (k, g)-graph.Let z 1.Then there exists a cyclic lift B α that contains a connected component G such that (i) G is k-regular, (ii) the girth of G is at least g, (iii) no 2 vertices of G that belong to the same fiber share mutual neighbors, and (iv) at least one fiber of G contains at least z vertices.
Proof.Let B be a k-regular graph of girth g and order n.Take m > z • n and relatively prime to g and let u be a vertex of B that lies on a g-cycle of B. Choose a direction for the g-cycle containing u and define a voltage assignment α from D(B) into Z m by assigning 1 to one of each pair of opposing darts of B and −1 to the opposite dart in such a way that the darts of the g-cycle containing u oriented along the direction chosen for this cycle all receive the voltage 1.Let (u, g) be any lift of u in B α , and let G be the connected component of (u, g).Then G satisfies the conditions of our lemma.First, each vertex of B α is of degree k, and so must be the degree of each vertex of G.
Consider a cycle C of B α of length .If C contains no two vertices from the same fiber of B α (other than the beginning and end vertex), the projection of C into B must be a cycle of B of length as well, and hence g.If C does contain at least two vertices of the same fiber, it contains a sub-path P whose end-points belong to the same fiber, but no other two vertices do.The projection of P into B cannot be a self-reversing path as the base graph B has no loops and no multiple edges and thus the projection is a cycle of B, and hence g again.It follows that the girth of G is at least g.Let v f , v h ∈ V (G), and suppose that there exists a vertex w r that is a mutual neighbor of v f and v h .Then there must exist an edge e of B that connects v and w.Suppose that the electronic journal of combinatorics 22(1) (2015), # P1.77 the voltage of the dart from v to w is a (where a is either 1 or −1).Since w r is a neighbor of v f , r = f + a.Similarly, since w r is a neighbor of v h , r = h + a.But f and h were assumed different, and we obtain a contradiction.
Finally, it is not hard to see that the order of G is at least m (the order of the voltage group).The argument is quite simple.Let C be the g-cycle through u whose edges have been assigned 1's in accordance with its direction.Then the net voltage of C is equal to g, and the length of each lift of C in B α is equal to g • |g| where |g| denotes the order of g in Z m .Since m is assumed to be relatively prime to g, the order of g in Z m is m and hence the lift of C passing through u g is a cycle of length gm.As all the vertices of C belong to G, the order of G is greater than m > z • n, where n is the number of fibers.By the Pigeonhole Principle, G contains at least z vertices of at least one fiber of B α .
6 Connected (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graphs We prove the existence of a connected (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graph for each set of parameters satisfying the condition k 1 3.It is once again the case that the first part of our proof, namely the part where we prove the existence of the (k 1 ; g 1 , g 2 , . . ., g s ; N )-graphs, can already be deduced from Theorem 1 (but we prove it differently).Graphs with varied degrees are then obtained by extending the initial (k 1 ; g 1 , g 2 , . . ., g s ; N )-graphs.
Proof.For t = 1 and k 1 = 3, let B be any connected (k , g )-graph with k , g > max{N, 3}, and let G be the connected component of a lift of B that we constructed in Lemma 9 with z = s.Then G is of degree k > 3 and girth at least g > N .Let v 1 , v 2 , . . ., v s be the s vertices that belong to the same fiber of B whose existence is guaranteed by Lemma 9 (iv).Then all their neighborhoods are mutually disjoint and each of the neighborhoods is of size k .For each of the numbers g j , 1 j t, decrease the degree of the vertex v j to g j (by removing the appropriate number of adjacent edges).The removal of these edges will result in a graph H with one vertex of degree g j for each 1 j s, and the rest of the vertices of degrees k −1 or k (depending on whether they had been a neighbor of a vertex whose degree has been decreased and the connecting edge has been removed or not).If ρ is any vertex-neighborhood labeling of H, for each 1 i s, C g i is a g i -cycle labeled at random from {1, . . ., g i }, and C k −1 , C k are randomly labeled k − 1 and k -cycles, then the generalized truncated graph T (H, ρ; C g 1 , C g 2 , . . ., C gs , C k −1 , C k ) can be easily seen to be a (3; g 1 , g 2 , . . ., g s ; N )-graph.(Since k is taken to be greater than N , the cycles C k −1 and C k do not have to be listed among the cycles shorter than N .) The proof for the cases t = 1 and k 1 > 3 can now be completed using the 'usual' recursion via inserting a (k − 1; g 1 , g 2 , . . ., g s ; N )-graph of order n into any (n, g)-graph with g > N to obtain a (k; g 1 , g 2 , . . ., g s ; N )-graph.
Consider finally the case t > 1 and k 1 3.In this case, we construct the (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graphs from the above (k 1 ; g 1 , g 2 , . . ., g s ; N )-graph G by increasing the degrees of some of the vertices of the graph G to the desired values k 2 , k 3 , . . ., k t .Let us the electronic journal of combinatorics 22(1) (2015), #P1.77 assume without loss of generality that G contains at least t−1 non-incident edges e 2 , . . ., e t with end-points x i , y i , 2 i t, respectively.Let H be any connected k 1 -regular graph of girth g > N + 1, let e be an edge of H with end-points x and y, and let H be the graph H with the edge e (but not the vertices x, y) removed.For every i, 2 i t, attach k i − k 1 disjoint copies of H to the vertices x i , y i with each of the copies attached via a pair of edges connecting x to x i and y to y i .We claim that the resulting graph is our desired (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graph: The degrees of all but the vertices x, y, x i , y i , 2 i t, remain unchanged (equal to k 1 ), the degrees the vertices of x, y in each copy of H equal k 1 − 1 + 1 = k 1 , and the degrees of the vertices x i , y i clearly equal k i , for all 2 i t.Thus, the resulting graph has the correct degree sequence.As G already contains cycles of all of the lengths g i , 1 i s, and no edges have been removed from G, the resulting graph contains all the required cycles.It remains to show that the resulting graph does not contain cycles of length smaller than N that are not among the required ones.As no edges have been added between any two vertices of G, and no edges have been added between any two vertices of H , if such a cycle existed, it would have to contain edges from both G and some copy of H , and thus would have to contain both vertices x and y of some copy of H .However, since the girth of H was taken to be greater than N + 1, any path between x and y that consists entirely of edges in H must be of length at least N as otherwise this path together with the (removed) edge between x and y would have given rise to a cycle of length smaller than N + 1.Thus, any cycle of the resulting graph that contains both x and y must be longer than N , and therefore the resulting graph does not contain cycles that contain edges from G and some copy of H and are shorter than N .This completes the proof of the theorem for this last case.
We feel obliged to note that the above proof is a simplification of our original proof.This has been achieved using ideas submitted by one of our referees.
7 Bipartite (k 1 , . . ., k t ; g 1 , . . ., g s ; N )-graphs and configurations As mentioned in the introduction, the motivation for considering bipartite graphs with these properties comes from the study of combinatorial configurations [4].
A (v k ) configuration is an incidence structure of v points and v lines satisfying the properties that each line is incident with k points, each point is incident with k lines, and any two distinct points are incident with at most one common line.The incidence graph (also called a Levi graph) of a (v k ) configuration is a bipartite graph with the vertices in one bipartition set representing the points, the vertices in the other bipartition set representing the lines, and the adjacency relation defined by the incidence relation in the configuration.From the definition it immediately follows that a Levi graph of a (v k ) configuration is a k-regular graph of girth at least 6.Conversely, each bipartite k-regular graph with girth at least 6 determines a pair of mutually dual (v k ) configurations.
A cycle of length g (which is an even number) in the Levi graph of a configuration C corresponds to a g 2 -lateral (or a g 2 -gon) in C. Formally, an n-lateral (or an n-gon) in a configuration is a cyclically ordered set {p 0 , 0 , p 1 , 1 , . . ., p n−2 , n−2 , p n−1 , n−1 } of pairwise the electronic journal of combinatorics 22(1) (2015), # P1.77 (k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N ) graph with a t-vertex coloring and the property that each vertex of color i has degree k i .Using this notation we no longer require that the k i be distinct (and ordered).The generalized girth problem has now a special subproblem for [k 1 , k 2 , . . ., k t ; g 1 , g 2 , . . ., g s ; N ]-graphs.
8 Vertex-transitive (k; g 1 , g 2 , . . ., g s ; N )-graphs A graph G is called vertex-transitive if for any ordered pair (u, v) of vertices of G there exists an automorphism φ of G such that φ(u) = v.As every vertex-transitive graph must be k-regular for some k, none of the non-regular graphs constructed so far in our paper are vertex-transitive.Thus, in this section, in the context of vertex-transitive graphs, we will only consider (k; g 1 , . . ., g s ; N )-graphs.In view of the vertex-transitive constructions in [19,15,9], it is interesting to ask whether we may be able to construct a vertex-transitive (k; g 1 , g 2 , . . ., g s ; N )-graph for any parameter set (k; g 1 , g 2 , . ., g s ; N ).To answer this question, we present a negative result that shows that for certain parameter sets there exist no vertex-transitive graphs.
Proof.Due to their high level of symmetry, vertex-transitive graphs have the same cycle structure through each vertex of the graph.Thus, if a trivalent vertex-transitive graph G contains a g 1 -and a g 2 -cycle, each vertex of G must lie on at least one g 1 -and at least one g 2 -cycle.Since G is trivalent, this means that there exist a g 1 -cycle C 1 and g 2 -cycle C 2 of G that share at least one edge.Let P 1 , P 2 , . . ., P r be the maximal subpaths shared by C 1 and C 2 of lengths 1 , 2 , . . ., r , respectively.
The assumption r = 1 leads to a contradiction.
If r > 1, consider the cycles obtained from the union of C 1 and C 2 after removing the paths P 1 , P 2 , . . ., P r .There is exactly r of them, say of lengths g 1 , g 2 , . . ., g r .Then and since each of the g i 's is either equal to g 1 or 2g 1 , and all the i 's are 1, there cannot be more than 2 such cycles and they both have to be of length g 1 .This is once again impossible, as Note that our non-existence result bears further insight into the question about the distribution of face lengths in embeddings of Cayley graphs addressed in [23] as well as in the constructions of vertex-transitive cages [15,9].It appears extremely likely that there are many further restrictions of the above type, and the classification of all the parameters sets (k; g 1 , g 2 , . . ., g s ; N ) that admit the existence of a vertex-transitive (k; g 1 , g 2 , . . ., g s ; N )-graph may prove to be very interesting (and complicated).a graph can share an edge as such an occurrence would lead to the existence of gcycle of length strictly between g and N .Let C be a g-cycle of G, assume that the vertices of C are the vertices v 1 , v 2 , . . ., v g , and consider the g(k − 2) edges incident to the vertices v i not included in C. We claim that each v i is attached to a separate 'tree' of (k /2 vertices (with any two trees mutually disjoint).The argument is similar to the argument used to prove the Moore bound.Consider the end-points of the (k − 2) edges adjacent to v i and not belonging to C. Each of these must be adjacent to k − 1 new vertices not shared with any other branch of the tree at v i and neither with any other tree rooted at v j , j = i.This is due to the fact that we cannot create shorter cycles than g and cannot even create cycles of length g that would share a part of C. The first time an edge from a tree from v i can go to another branch of that tree or to one of the other trees is at the distance (g − 1)/2 from v i .This completes the proof for odd g.
The case for even g differs from the odd case in that there might exist two cycles of length g that share at least an edge.More precisely, if g = 2m, there might exist two g-cycles that share an m-path.If no such two g-cycles exist in G, a very similar proof to the odd case shows the bound stated.If G contains two g-cycles sharing an m-path, trees attached to vertices antipodal with respect to these cycles (i.e., lying on the same cycle and of distance m) may form another m-path and thus the attached trees may share vertices.It is not hard to see, however, that this is not possible.Namely, if there existed two distinct vertices anywhere on these two g-cycles that were joined by an additional path of length m, one could always construct a cycle involving this path and parts of the original g-cycles of length smaller than 2g but not equal to g.Thus, all the trees attached to the 3m vertices of the two g-cycles must be disjoint in this case as well, and the number of vertices of the graph G must be at least 3m ( where the second bound could clearly be improved.Both upper bounds are obtained by considering disconnected graphs which yields another interesting aspect of the problem of the minimal order of an (k; g 1 , g 2 , . . ., g s ; N )-graph that does not occur with regard to (k, g)-cages: the order n(k; g 1 , g 2 , . . ., g s ; N ) can be achieved by a disconnected union of smaller graphs (as is the case for k = 2).The smallest such example for k = 3 is the (3; 3, 4, 6; 8)-cage which turns out to be a union of the complete graph K 4 , the smallest (3; 3, 4; 8)-graph, and the complete bipartite graph K 3,3 , the smallest (3; 4, 6; 8)-graph.It has order 10 while the smallest connected (3; 3, 4, 6; 8)-graph has order 14.It is also possible that n(k; g 1 , g 2 , . . ., g s ; N ) is achieved by both a connected and a disconnected graph.The smallest case for k = 3 is n(3; 4, 5, 6; 8): there exist connected and disconnected (3; 4, 5, 6; 8)-graphs of order 16 which is the smallest value for this set of parameters.On the other hand, there exists an infinite family of triples (g i , g i ; N i ), g i < g i < N i and k 3, such that n(k; g i , g i ; N i ) < n(k; g i ; N i ) + n(k; g i ; N i ), and the corresponding generalized cage is connected.A somewhat disappointing example of this situation can be obtained as follows: where N is assumed to be > 4. At this point, we do not have a more meaningful example of this situation.One of the fundamental problems when trying to determine a precise value of n(k, g) (the order of the smallest (k, g)-graph) is that beside having to construct a graph of order n(k, g), one also has to establish the non-existence of a smaller (k, g)-graph.Although obviously one is to expect the same problem to occur with regard to our generalized cages, there are certain parameter sets where establishing the non-existence of smaller graphs of those parameters proves out to be a bit less demanding.One such example are the sets of parameters with k = 3 and the required cycle lengths covering the whole range of cycle lengths from the girth 3 through N − 1 -the (3; 3, 4, . . ., N − 1; N )-cages.These are depicted in Figure 4 for N 4 and it is easy to see that the graphs G N have cycles of all the lengths 3, 4, . . ., N − 1 while being of the smallest possible orders (N for N even and the electronic journal of combinatorics 22(1) (2015), #P1.77N − 1 for N odd) -due to the fact that they must contain an N − 1 cycle.These graphs can be thought of as in some sense opposite to the graphs considered in Lemma 14.
To conclude our paper, we point out that the above cages are not unique with respect their parameters.It is easy to see that the graph depicted in Figure 5 is also a (3; 3, 4, 5, 6, 7, 8; 9)-graph but it is not isomorphic to the graph G 8 = G 9 .

Figure 3 :
Figure 3: Truncation of a vertex of degree 4 by a 4-cycle with a chord The set of vertices {u g | g ∈ Γ} is called the fiber of u.If C = e 1 , e 2 , . . ., e is an oriented cycle of B, the product f = α(e 1 ) • α(e 2 ) • • • • • α(e ) (in Γ) is called the net voltage of C, and a lift of C is a cycle of length equal to the lcm( , |f |), where |f | denotes the order of f in Γ.