On Mixed Cages

Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.


Introduction
The cage problem, which asks for the identification of smallest regular graphs of given girth, has been extensively studied for graphs and for digraphs.A series of survey papers have tracked progress on the problem for graphs Wong (1982); Biggs (1998); Exoo and Jajcay (2013).The digraph version of the problem was first considered in 1970 by Behzad, Chartrand and Wall Behzad et al. (1970).Recently, Araujo-Pardo, Hernández-Cruz, and Montellano-Ballesteros Araujo-Pardo et al. (2019) considered the problem for mixed graphs and raised a number of interesting questions.
Finding graphs that give reasonable upper bounds and proving lower bounds both seem to be difficult problems for the undirected case.The situation appears to be different for digraphs.It is generally believed that the extremal digraphs were correctly identified by Behzad, Chartrand and Wall Behzad et al. (1970), who proposed a fundamental conjecture discussed below.For mixed graphs, as suggested in Araujo-Pardo et al. (2019), the level of difficulty in finding good candidate graphs may depend on the relative sizes of the directed and undirected degrees.
In this paper, we address and resolve some of the questions raised in Araujo-Pardo et al. (2019).In particular, it is shown below that the Moore-like bound they established for directed degree 1 can be achieved.In the next section, we review basic notation and terminology.Then we review the basic bounds of Moore, of Behzad, Chartrand and Wall, and of Araujo-Pardo, Hernández-Cruz, and Montellano-Ballesteros.Finally, we present a number of new results.

Geoffrey Exoo
in-degree and out-degree of v, denoted ideg(v, G) and odeg(v, G), are the numbers of arcs incident to and from v. A mixed graph G is regular if the degree and out-degree are constant as v ranges over the vertices of G. If, in addition, the in-degree is constant then the graph is totally regular.
A cycle in a mixed graph is a sequence of vertices v 0 , v 1 , • • • , v k such that v 0 = v k , and each pair of consecutive vertices (v i , v i+1 ) is either joined by an edge or an arc (directed from v i to v i+1 ), and there are no repeated edges or arcs.The girth of a mixed graph is the length of a shortest cycle.Note that the definition considers the possibility of 1-cycles (loops) and 2-cycles.As usual, we use C k to denote an undirected k-cycle, and C k to denote a directed k-cycle.
An (r, z, g)-graph (i) is a regular mixed graph with degree r, out-degree z, and girth g.We are interested in cases where r, z > 0 and g ≥ 5.An (r, z, g)-cage is an (r, z, g)-graph of minimum possible order.We denote this minimum order by f (r, z, g).
For graphs or digraphs G and H, the graph composition Harary (1969).This is also known as the lexicographic product.
Finally, we note that modular arithmetic on subscripts will frequently be employed when specifying constructions.In obvious cases, we will try to avoid cluttering the presentation with frequent reminders.In one case we use expressions of the form a + b mod c and remind the reader that the modulus operator has the precedence of multiplication.

Basic Bounds
The following conjecture is fundamental to the cage problem for directed graphs.
Behzad, Chartrand and Wall also identified the digraphs, for each r and g, that they believed to be extremal, These digraphs, which we denoted BCW (r, g), have orders r(g − 1) + 1 with vertices v i for 0 ≤ i ≤ r(g − 1) and arcs v i v i+j for 1 ≤ j ≤ r.A small example is shown in Figure 1.
In Araujo-Pardo et al. (2019) Araujo-Pardo, Hernández-Cruz, and Montellano-Ballesteros considered the problem of finding mixed cages.They focused on the case z = 1 and found a lower bound for f (r, 1, g) based on the well known Moore bound for undirected graphs.Their idea is to attach undirected Moore trees to each vertex of a directed path of order g, choosing trees whose depth is as large as possible while still guaranteeing that all tree vertices are distinct.
Recall that the Moore bound for an r-regular graph of diameter d is given by: be the vertices of a path of order g.Attach a Moore tree of depth i to v i and v g−1−i (if distinct), for 0 ≤ i ≤ ⌊g/2⌋.This gives the following bound.
A small example AHM tree is shown in Figure 2. We will see that this bound can, at least occasionally, be achieved.A central question for mixed cages is to determine how often it is achieved.Upper bounds on f (r, 1, g) can be obtained from known cages (or candidates) of degree r + 2 and girth g whenever the graph is Hamiltonian (or at least has a 2-factor with no short cycles).We shall refer to the upper bound obtained this way as the Cage Bound.In general, the Cage Bound is probably weak, but in at least one case it may be useful.The Hoffman-Singleton graph is the (7, 5)-cage, and is Hamiltonian.If any Hamiltonian cycle is cyclically oriented, a (5, 1, 5) mixed graph of order 50 is obtained.
It appears that most cages are Hamiltonian.A theorem of Singer Singer (1938) on collineations implies that girth 6 cages constructed from field planes are Hamiltonian.More recently, Lazebnik, Mellinger and Vega Lazebnik et al. (2013) showed that the incidence graphs of all finite projective planes are Hamiltonian.This author has tested all the remaining cages and cage candidates listed in Table 6 of the Dynamic Cage Survey Exoo and Jajcay (2013) that are relevant to the results in this paper and found them all to be Hamiltonian.The cage bound may at least serve as a useful benchmark.One might view finding (r, 1, g)-graphs whose orders are less that the orders of the (r+2, g)-cages (or candidates) as significant first steps.
4 The Case r + z ≤ 3 In this section, mixed cages for r, z ≤ 2 are determined.The first theorem uses what could be viewed as a degenerate case of the AHM bound, as it employs Moore trees of degree one, which are single edges.
f (1, 1, g) = 2g − 2 Proof: The lower bound follows from the AHM bound.The AHM bound is achieved by the Möbius ladder, where the outer cycle (see Figure 3) is cyclically oriented.
Fig. 3: The Möbius ladder of order 8: the (1, 1, 5)-cage.An AHM tree can be found by using any directed path of length four and the three edges incident with the non-endvertices of the path.
Theorem 3 (AHM Araujo-Pardo et al. ( 2019)).The value of f (1, 2, g) can be determined using the r = 2 case of Behzad-Chartrand-Wall Conjecture, proved by Behzad, and by the graph Proof: Let G be a (1, 2, g)-graph.Consider the directed subgraph.The result of Behzad Behzad (1973) on the case r = 2 of the Behzad-Chartrand-Wall Conjecture implies that G must have at least 2g − 1 vertices.But since r is odd, the order of G must be even.So 5 The Case r = 2 In this section we present three theorems that provide upper bounds for f (2, z, g).When r = 2, a simple upper bound can be obtained using subgraphs of Theorem 5. f (2, z, g) ≤ g 2 , for z ≤ g.Proof: The required graphs are all subgraphs of C g [C g ].Construct a (2, z, g)-graph G as follows.The vertices of G are {v i,j | 0 ≤ i, j < g}, with edges v i,j v i,j+1 , and arcs v i,j v i+1,j+k for 0 ≤ k < z (subscript arithmetic is modulo g).
For a restricted range of z, we can remove one of the g-cycles and still maintain the girth.
f (2, z, g) =    g 2 − g, g even and z ≤ g/2 g 2 − 1, g odd and z ≤ (g+1)/2 Proof: If g is even, let g = 2h, and then z ≤ h.Construct a (2, z, g)-graph G of order g 2 − g as follows. Let The arc set is a bit more complex: A small example of the construction in Theorem 6 is shown in Figure 5. Finally, when r = z = 2 we reduce the coefficient of g 2 in the upper bound to 3/4.
g even, On Mixed cages 7 Proof: Let s = g 2 , and t = 3g 2 .We describe a graph G of order st with r = z = 2, and girth g.Let The undirected subgraph of G consists of s disjoint t-cycles, where the first subscript of a vertex indicates which cycle the vertex is on, and the second subscript indicates its position on the cycle.We imagine the cycles are numbered according to the first coordinate their vertices, and refer to cycle 0, cycle 1, on up to cycle s − 1, with the vertices of each cycle numbered in the natural order.
Next we specify the arcs so that all arcs from vertices on cycle i go to vertices on cycle i Finally the arcs from cycle s−1 back to cycle 0: Note that the function that maps v i,j to v i,j+1 is a graph automorphism.Any possible cycle of length less than g must contain arcs, since the undirected subgraph consists of disjoint t-cycles.But any cycle containing an arc contains at least t arcs and at least one arc joining each pair of consecutive t-cycles.Hence it also contains at least one vertex from each of the undirected cycles.So by symmetry, if there is a cycle of length less than g there is such a cycle containing vertex 0. But any such cycle must contain exactly r arcs and fewer than g − r = ⌈ g 2 ⌉ edges.By construction, this is impossible.
Note that if this construction were modified for f (2, 3, g) we would need undirected cycles of length 2g and would no longer have an improvement over the upper bound in Theorem 5.
For the case (2, 2, 5), a graph of order 19 was found by Claudia De La Cruz and Miguel Pizaña (personal communication).Their graph can viewed as a mixed orientation of the cubic residue graph.The graph in Figure 6 is a smallest (2, 2, 6)-graph.As noted in Araujo-Pardo et al. (2019), the cases where z = 1 may result in graphs most similar to undirected cages.In these cases the directed subgraph consists of a set of disjoint cycles whose lengths partition the order of the graph.
Theorem 8. f(3,1,5) = 24 Proof: The AHM bound in this case is 20, so one needs to show that (3, 1, 5)-graphs of orders 20 and 22 do not exist.This was done by exhaustive computer searches, which made heavy use of the nauty and Traces McKay and Piperno (2014) package.The searches were completed by two different programs.
The first program breaks the task into cases based on the structure of the (maximal) directed subgraph of a putative (3, 1, 5)-graph.This subgraph must be a union of cycles.The lengths of these cycles give a partition of 20 (respectively 22) with no part less than 5.There are 13 (resp.18) such partitions.For each case, the program then does an exhaustive search for an undirected subgraph.
The other program reverses the procedure, and divides the search into cases according to the structure of the undirected subgraph.This subgraph must be a cubic graph whose girth is at least 5.There are 5784 such graphs of order 20 and 90940 of order 22 McKay and Piperno (2014).While this leaves more cases to consider than the first program, the exhaustive search for each case is much faster.
After completing the searches for n = 20 and n = 22, and finding no graphs, the first of these methods was applied to the case n = 24, and 23 different graphs were generated.The most symmetric of these graphs is depicted in Figure 7. Fig. 7: A graph of order 24 with (r, z, g) = (3, 1, 5).There are edges joining inner vertices to all outer vertices of the same color.Fig. 8: One of the two cubic graphs of order 12 with girth 5.Each of the three inner vertices is adjacent to the three vertices on the 9-cycle having the matching color.
The undirected subgraph in Figure 7 is isomorphic to two copies of this graph.
The undirected subgraph of the graph in Figure 7 is isomorphic to two copies of the cubic graph of girth In all cases, addition of subscripts is done modulo 10.
The graph described in the above proof is depicted in Figure 10.
Next we present a few constructions where we were able to come relatively close to the AHM bound.
All lower bounds are derived from the AHM bound.The upper bound constructions are handled separately.
(a) f (3, 1, 7) ≤ 60 The directed subgraph for the (3, 1, 7)-graph consists of 6 disjoint directed 10-cycles.Each of these cycles represented by a node in Figure 11.It will be useful to imagine the 60 vertices of the graph labeled v i,j for 0 ≤ i < 6 and 0 ≤ j < 10.In this way, node i in the figure represents all vertices with first subscript i.Each these sets of 10 vertices induces a directed 10-cycle labeled in the natural order.
Various pairs of 10-cycles are connected by undirected matchings.The arcs in the figure do not represent arcs in the graph, but are there to specify the matching between pairs of 10-cycles.For example, the arc from node 1 to node 2 is labeled with a 4.This means there is a matching connecting 10-cycle number 1 to 10-cycle number 2 such that vertex i in the first 10-cycle is adjacent to vertex i + 4 in the second 10-cycle, i.e., v 1,i is adjacent to v 2,i+4 , and again subscript addition is modulo 10.
The reader familiar with voltage graphs can think of the arc labels as voltage assignments from the cyclic group of order 10.
(b) f (3, 1, 8) ≤ 76 Construct a (3, 1, 8)-graph G on vertices v i,j for 0 ≤ i < 2 and 0 ≤ j < 38.For fixed i, the 38 vertices v i,j form a directed cycle, labeled in the natural order.The edge set of G is comprised of edges of one of the following forms: The automorphism group of G has order 38, with each directed 38-cycle comprising an orbit.(c) f (4, 1, 5) ≤ 34 An example (4, 1, 5)-graph of order 34 is shown in Figure 12.The directed subgraph consists of two 17-cycles, as shown in the right half of the figure.The undirected subgraph is shown in the left half of the figure.We identify the vertices in the left figure with the corresponding vertices in the right figure.Label the vertices v i,j for 0 ≤ i < 2 and 0 ≤ j < 17, where we imagine the vertices with first subscript 0 are those on the outer circle.Then there are arcs from v 0,i to v 0,i+7 and from v 1,i to v 1,i+6 , and edges v 0,j v 1,j−2 and v 0,j v 1,j+2 .
g even The extremal graphs in Araujo-Pardo et al. (2019) are derived from the Behzad, Chartrand and Wall digraphs by replacing one directed spanning cycle by an undirected cycle.The lower bound follows from the AHM bound, giving the first examples where the bound is achieved.

Fig. 6 :
Fig. 6: A (2, 2, 6)-graph of order 27.Only one pair of arcs from the green cycle to the blue cycle are shown.The others are obtained by rotation.

Fig. 11 :
Fig. 10: The unique smallest (3, 1, 6)-graph of order 30.Vertices in the lower figure are adjacent to vertices in the upper figure that have the same color.

Fig. 12 :Fig. 13 :Fig. 14 :
Fig. 12: A (4, 1, 5)-graph of order 34.The figure on the left shows the edges and the figure on the right shows the arcs.