Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7 and order 2q 3 + q 2 + 2q for each even prime power q 4, and of order 2q 3 + 2q 2 − q + 1 for each odd prime power q 5.


Introduction
Throughout this paper, only undirected simple graphs without loops or multiple edges are considered.For terminology and notation not explicitly defined here, please refer to [14].Let G be a graph with vertex set V = V (G) and edge set E = E(G).We denote the subgraph of G induced by a subset U ⊂ V (G) as G[U ], and it is the graph with V (G[U ]) = U and for any u, v ∈ V (G[U ]) the edge uv belongs to E(G[U ]) if and only if uv ∈ E(G).The girth of a graph G is the number g = g(G) of edges in a smallest cycle.For every v ∈ V , N G (v) denotes the neighbourhood of v, that is, the set of all vertices adjacent to v, we may denote it simply by N (v).Similarly, for each positive integer t we denote by N t (v) the neighborhood of v at distance t, i.e. the set N t (v) = {x ∈ V (G) : d(x, v) = t}, and the neighborhood of an edge uv at distance t is the set N t (uv) = {x ∈ V (G) : d(x, u) = t or d(x, v) = t}.
The degree of a vertex v ∈ V is the cardinality of N (v).A graph is called regular if all the vertices have the same degree.A (k, g)-graph is a k-regular graph with girth g.Erdős and Sachs [16] proved the existence of (k, g)-graphs for all values of k and g provided that k 2. Thus most work carried out has focused on constructing a smallest one [1,3,4,5,6,7,8,9,10,11,15,17,19,20,22,23,27,28,31].A (k, g)-cage is a kregular graph with girth g having the smallest possible number of vertices n(k, g).Cages have been studied intensely since they were introduced by Tutte [35] in 1947, and their construction is a very difficult task.
Counting the numbers of vertices in the distance partition with respect to a vertex yields Moore's lower bound n 0 (k, g) (cf.e.g.[18,Eq.(2)]) with the precise form of the bound depending on whether g is even or odd: if g is even. (1) Biggs [12] calls the excess of a (k, g)-graph G the difference |V (G)| − n 0 (k, g).The construction of graphs with small excess is also quite challenging.Biggs is the author of a report on distinct methods for constructing cubic cages [13].More details about constructions of cages can be found in the survey by Wong [39] or in the book by Holton and Sheehan [21] or in the more recent dynamic cage survey by Exoo and Jajcay [18].
A (k, g)-cage with n 0 (k, g) vertices and even girth exists only when g ∈ {4, 6, 8, 12} [19].If g = 4 they are the complete bipartite graph K k,k , and for g = 6, 8, 12 these graphs are the incidence graphs of generalized g/2-gons of order k −1.This is the main reason for (k, g)-cages with n 0 (k, g) vertices and even girth g are called generalized polygon graphs [12].In particular a 3-gon of order k − 1 is also known as a projective plane of order k − 1.The 4-gons of order k − 1 are called generalized quadrangles of order k − 1, and, the 6-gons of order k − 1, generalized hexagons of order k − 1.All these objects are known to exist for all prime power values of k − 1, and no example is known when k − 1 is not a prime power (cf.e.g.[30, p.25], [34]).
In this paper, we construct new infinite families of small regular graphs of girth 7. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power, which are summarized in Section 2. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs (cf.Definitions 2 and 9).In Section 3 we construct (q + 1, 7)-graphs of order 2q 3 + q 2 + 2q for each even prime power q 4 (cf.Construction 1 and Theorem 7).In Section 4 we construct (q + 1, 7)-graphs of order 2q 3 + 2q 2 − q + 1 for each odd prime power q 5 (cf.Construction 2 and Theorem 15).All these graphs are the smallest (q + 1, 7)-graphs known so far, for each prime power q > 5.

Preliminaries
It is well known [24,30] that Q(4, q) and W (3, q) are the only two classical generalized quadrangles with parameters s = t = q.
For any generalized quadrangle Q of order (s, t) and every point x of Q, let x ⊥ denote the set of all points collinear with x.For a nonempty set X of vertices of Q, we define X ⊥ := x∈X x ⊥ .Note that N 2 (x) in the incidence graph Γ q , corresponds in the geometry to x ⊥ for a point x ∈ Q.
The span of the pair (x, y) is sp(x, y) = {x, y} ⊥⊥ = {u ∈ P : u ∈ z ⊥ ∀ z ∈ x ⊥ ∩ y ⊥ }, where P denotes the set of points in Q.If x and y are not collinear, then {x, y} ⊥⊥ is also called the hyperbolic line through x and y.If the hyperbolic line through two noncollinear points x and y contains precisely t + 1 points, then the pair (x, y) is called regular.A point x is called regular if the pair (x, y) is regular for every point y not collinear with x.It is important to recall that the concept of being regular also exists for a graph.Hence, we will emphasize when the word "regular" refers to a point of a geometry or to a graph.Remark 1. [30, p.33, dual of 3.3.1(i)]Every point in W (q) is regular (i.e.|sp(x, y)| = q + 1 for all non-collinear x, y).
There are several equivalent coordinatizations of these generalized quadrangles (cf.[29], [36], [37], see also [24]) each giving a labeling for the graph Γ q .In Section 4 we present a further labeling of Γ q , equivalent to previous ones (cf.[1,2]), which will be central for our constructions since it allows us to keep track of the properties (such as regularity and girth) of the small regular graphs of girth 7 obtained from Γ q .
3 Construction of small (q + 1, 7)-graphs for even prime powers In this section we construct a family of (q + 1, 7)-graphs of order 2q 3 + q 2 + 2q obtained from a (q + 1, 8)-cage Γ q for each even prime power q 4. In general terms, we proceed by removing, from a (q + 1, 8)-cage, a subgraph H consisting of a distinguished vertex x, its neighbours, and almost all its second neighbours (the neighbourhoods of all but two of the neighbours of x).The resulting graph is not regular, indeed the neighbours of the subgraph H in the cage, are left with degree q.So we add appropriate matchings among such vertices to restore the (q + 1)-regularity of the graph.The constructed graph has girth at most 7 by Equation (1).The details in this Section are devoted to choosing the matchings in an appropriate way to obtain girth exactly 7 in the graph.
Definition 2. Let Γ q be a (q + 1, 8)-cage for an even prime power q 4 and H as in (4).We define Γ 1 q to be the graph with: Remark 3. The graph Γ 1 q has order |V (Γ q )| − (q 2 + 2) and all its vertices have degree q + 1.Furthermore, the girth of Γ 1 q is at most 7 by Equation ( 1).Remark 4. Let u and v be distinct vertices of a graph G of girth 8 such that there is a uv-path P of length t < 8. Then every uv-path P such that E(P ) ∩ E(P ) = ∅ has length the electronic journal of combinatorics 22(3) (2015), #P3.5 Proposition 5. Let Γ q be a (q + 1, 8)-cage for an even prime power q 4 and Γ 1 q as in Definition 2. Then Γ 1 q has girth 7 if the following condition holds: For each uv ∈ M X ij and each X kl , where i, k ∈ {0, . . ., q − 2}, j, l ∈ {1, . . ., q} Proof.By Remark 3 the graph Γ 1 q has girth at most 7. From Remark 4, the distances in Γ q − H between the elements in the sets Z ∈ Z satisfy the following: 3, for i ∈ {0, 1}, k ∈ {2, . . ., q} and j ∈ {1, . . ., q}.
• If e 1 ∈ M X tj and e 2 ∈ M X lk , then |C| 7 for t = l, by condition (*).
Figure 1: The dashed edges and their lower end-vertices illustrate the subgraph H, removed from Γ q , when q = 4. Above, find the choice of the matchings that lead to a (5, 7)-graph of girth exactly 7 and order 152.
Every vertex w jh is adjacent to exactly one vertex in X ij since the girth of Γ q is 8. Denote {x ijh } = N (X ij ) ∩ N (w jh ) for each i ∈ {2, . . ., q} and j, h ∈ {1, . . ., q} (see Figure 1).Note that x ijh is well labeled, because if x ijh had two neighbors w jh , w jh ∈ q i=2 N (X ij ), then Γ q would contain the cycle x ijh w jh x i jh x i j x i jh w jh of length 6.
To prove that the matchings M X ij defined in this way fulfill condition (*), suppose that x ijh x ijh ∈ M X ij and x i jh x i jh ∈ M X i j for i = i.Then F i and F i would have the edge hh in common contradicting that they come from a 1-factorization.
To conclude, notice that for uv ∈ M X ij and a, b ∈ X kl with l = j and possibly k = i, the distances d(u, a) and d(v, b) are at least 4.
Therefore, there exist q 2 − q matchings M X ij with the desired property.
Construction 1: Let q 4 be an even prime power.Let Γ 1 q be the (q + 1)-regular graph of order 2q 3 + q 2 + 2q from Definition 2 with M X ij as in the proof of Lemma 6, for i ∈ {2, . . ., q} and j ∈ {1, . . ., q}; and with M X 0 and M X 1 matchings of X 0 and X 1 respectively, chosen arbitrarily.Then, the graph Γ 1  q obtained with such a choice of matchings has girth 7 by Proposition 5.
As a consequence we have the following theorem.
Theorem 7. Let q 4 be an even prime power.Then, there is a (q + 1)-regular graph of girth 7 and order 2q 3 + q 2 + 2q.
Figure 1 illustrates this construction for q = 4.Note that this (5, 7)-graph has 152 vertices, as the two found in 2001 by McKay and Yang [26,32].
In this section we construct an infinite family of (q + 1, 7)-graphs of order 2q 3 + 2q 2 − q + 1 for odd prime power q 5. Analogously to Section 3, we will delete a set H of vertices from a (q + 1, 8)-cage Γ q and add matchings M Z between the remaining neighbors of H to obtain a small regular graph of girth 7.In general terms, the subgraph H consists of two distinguished vertices x and y at distance 4 in Γ q ; their neighbours and all but three of the common second neighbours of x and y (see Figure 2).The removal of H from Γ q leaves a non regular graph.Thus, we add appropriate matchings among the vertices of lesser degree, and three sporadic 2-paths, to restore the (q + 1)-regularity of the graph (cf.Definition 9).The constructed graph has girth at most 7 by Equation ( 1).As in Section 3, the details that follow are devoted to choosing the matchings in an appropriate way to obtain girth exactly 7 in the graph.In particular, some of the matchings can be chosen combinatorially, as for the even case (cf.Lemma 12).However, for the remaining ones, we rely on an algebraic coordinatization of the (q + 1, 8)-cage (cf.Definition 13 and Lemma 14).Specifically, the set H and matchings M Z , are defined as follows.
Figure 2: Subgraph of Γ q used to define H and Z.The subgraph H is highlighted by the dashed line.
Definition 8. Let x, y ∈ V (Γ q ) be vertices at distance 4 in Γ q , and let xx i s i y i y be the internally (vertex) disjoint xy-paths for i = 0, . . ., q (which exist since Γ q is (q + 1)connected, see [25]).We define the following sets (see Figure 2): Notice that the vertices of Γ q − H have degrees q − 1, q and q + 1.The vertices s 0 , s 1 , s 2 have degree q − 1, those in X i ∪ Y i ∪ S i have degree q and all the remaining vertices of Γ q − H have degree q + 1.Therefore, in order to obtain a (q + 1)-regular graph, we need to add edges to Γ q − H in such a way that cycles of length smaller than 7 are avoided.
Similarly as before, let Z be the family of all the sets X i , Y i , S i .Note that, all sets in Z have even cardinality.For each Z ∈ Z, M Z will denote a perfect matching of V (Z).Definition 9. Let Γ q be a (q + 1, 8)-cage for odd prime power q 5.
• Let Γ 1 q be the graph with: q and they are replaced by the paths of length two All vertices in Γ 1 q have degree q + 1 except for s 0 , s 1 , s 2 which remain of degree q − 1.Hence, by Definition 9, in Γ 2 q all vertices are left with degree q + 1.From Equation (1) the girth of both Γ 1 q and Γ 2 q is at most 7.
(a) Γ 1 q has girth 7 if the matchings M S i , M X i and M Y i have the following properties: (b) If both conditions (a1) and (a2) hold, the graph Γ 2 q also has girth 7.
Proof.By Remark 10 the graphs Γ 1 q and Γ 2 q have girth at most 7. From Remark 4, the distances in Γ q − H between the elements in the sets Z ∈ Z satisfy the following (see Figure 2): the electronic journal of combinatorics 22(3) (2015), #P3.5 Let C be a shortest cycle in Γ 1 q .If E(C) ⊂ E(Γ q − H), then |C| 8. Suppose C contains edges in M = Z∈Z M Z .If C contains exactly one such edge, then by (i), |C| 7. If C contains exactly two edges e 1 , e 2 ∈ M , the following cases arise: • If both e 1 , e 2 lie in the same M Z , then by (i), |C| 14 > 7.
• If e 1 ∈ M X i and e 2 ∈ M X j for i = j, by (ii), |C| 10 > 7.
• If e 1 ∈ M Y i and e 2 ∈ M Y j for i = j, by (iii), |C| 10 > 7.
• If e 1 ∈ M S i and e 2 ∈ M X j ∪ M Y j , by (v), |C| 8 > 7.
• If e 1 ∈ M S i and e 2 ∈ M S j for i = j, by item (a1), |C| 7.
q has girth 7, concluding the proof of (a).To prove (b), let C be a shortest cycle in Γ q with one vertex less than C, therefore |C| 8.
• If C contains two edges s i u i , s j u j , for i = j, their distances d Γ 1 q (s i , u j ) 4, d Γ 1 q (s i , s j ) 4, and d Γ 1 q (u i , u j ) 4. Since in either case C has length at least 7 and by Remark 10, the result holds.
The following lemma states the existence of the matchings M S i for the sets S i , which fulfill condition (a1) from Proposition 11.Notice that in the incidence graph of a general- and N (S i )| = q − 1, then the condition for the following lemma holds.
Proof.From the regularity of W (q) we know that q i=0 N (S i ) = {w 1 , . . ., w q−1 }, and since S i has q−1 vertices, every vertex w j is adjacent to exactly one vertex in s ij ∈ S i .Moreover, note that s ij is well labeled, because if s ij had two neighbors w j , w j ∈ q i=0 N (S i ), Γ q would contain the cycle (s ij w j s kj s k s kj w j ) of length 6.
Therefore, take the complete graph K q−1 , label its vertices as j ∈ {1, . . ., q − 1}.We know that it has a 1-factorization with q − 2 factors F 1 , . . ., F q−2 since q − 1 is even.For each i = 3, . . ., q + 1, let To prove that the matchings M S i defined in this way fulfill the desired property suppose that s ij s il ∈ M S i and s i j s i l ∈ M S i for i = i.Then F i and F i would have the edge jl in common contradicting that they were a factorization.So far, our construction has been independent from the coordinatization of the chosen (q + 1, 8)-cage, however, in order to define M X i and M Y i satisfying condition (a2) of Proposition 11, we need to fix all the elements chosen so far.To this purpose we use the following convenient description of a (q + 1, 8)-cage.Definition 13. [1, 2] Let F q be a finite field with q 2 a prime power and a symbol not belonging to F q .Let Γ q = Γ q [V 0 , V 1 ] be a bipartite graph with vertex sets

and edge set defined as follows:
For all a ∈ F q ∪ { } and for all b, c ∈ F q :

Or equivalently
For all i ∈ F q ∪ { } and for all j, k ∈ F q : Lemma 14.There exist matchings M X i and M Y i , for i = 0, . . ., q, such that condition (a2) in Proposition 11 holds.
We will distinguish two cases, when q = p is a prime or when q = p a , a > 1 is a prime power.Case 1: q = p a prime.
Recall that the vertices in Y i are in N ((i, 0, 0) 0 ) = N (y i ), the vertices in X j have coordinates ( , , j) 1 for = 1, . . ., p − 1.We will show that there exists a unique vertex and to this purpose we describe the coordinates of the vertices in X p in terms of the subscripts of w i j as ( , , s i j ) 1 where s i j ∈ {1, . . ., p − 1} and where the relationship between w i j and s i j is highlighted in what follows: Note that w i j = (a, b, c) 1 ∈ N ((i, 0, 0) 0 ) ∩ N 2 (( , , j) 1 ) ∩ N 2 (( , , s i j ) 1 ).
By equation (5) we have s i(− )j = −s i j implying, for fixed i and j, that the vertices w i j = (s i j , −is i j , is 2 i j ) 1 and w i− j = (−s i j , is i j , is 2 i j ) 1 in Y i are at distance two, respectively, only from the vertices ( , , j) 1 and ( , − , j) 1 in X j for j = {1, . . ., p}; and to the vertices ( , , ) and ( , , − ) in X p .Therefore the matchings M Y i and M X j for i ∈ {1, . . ., p − 1} and j ∈ {1, . . ., p} satisfy property (a2).
In order to define the matchings M X i and M Y i we proceed as above, but in this case we obtain the following equations: α s i j (α j−1 − α i−1 ) = α for i, j 1; α s 0 j (α j−1 ) = α for i = 0; α s i 0 (−α i−1 ) = α for j = 0.

Figure 4:
The matchings M X i and M Y i for i = 0, . . ., q − 1 and q = 9, determined by the 2 nd and the 1 rst coordinates of the vertices in X i and Y i respectively.
On the other hand, we obtain that s i +1j = s i j + 1, multiplying the equation ( 6) by α, which implies that the matchings M X i and M Y i satisfy property (a2), because the two vertices (α s i j , −α i−1+s i j , α i−1+2s i j ) 1 and (α s i j +1 , −α i−1+(s i j +1) , α i−1+2(s i j +1) ) 1 in Y i , are at distance two in X j only from the vertices ( , α , j) 1 and ( , α +1 , j) 1 , concluding the proof.
Construction 2: For q 5 an odd prime power let Γ 2 q be the graph of order 2q 3 + 2q 2 − q + 1 given in Definition 9, with the choice of matchings as in Lemmas 12 and 14.Then, the graph Γ 2 q is a (q + 1)-regular graph of girth 7 with 2q 3 + 2q 2 − q + 1 vertices as we prove in the following theorem.
exactly one such edge, then by (1), |C| 7. If C contains exactly two edges e 1 , e 2 ∈ M , we have the following cases: • If both e 1 , e 2 lie in the same M Z , then by (i), |C| 14 > 7.

Figure 3 :
Figure 3: The matchings M X i and M Y i for i = 0, . . ., p − 1 and p = 7, determined by the 2 nd and the 1 rst coordinates of the vertices in X i and Y i respectively.