Article No. eujc.1999.0335Available online at http://www.idealibrary.com on

Europ. J. Combinatorics (1999) 20, 789796

Strongly Closed Subgraphs in a Regular Thick Near PolygonAKIRA HIRAKI

In this paper we show that a regular thick near polygon has a tower of regular thick near sub-polygons as strongly closed subgraphs if the diameter d is greater than the numerical girth g.

c 1999 Academic Press

1. INTRODUCTION

Brouwer and Wilbrink [3] studied a regular thick near polygon of the numerical girth g = 4and showed the existence of a tower of regular thick near sub-polygons.

On the other hand we gave a constructing method of strongly closed subgraphs in a distance-regular graph of arbitrary numerical girth [6].

The purpose of this paper is to apply this constructing method to regular thick near polygonsof arbitrary numerical girth and to show the existence of a tower of regular thick near sub-polygons as strongly closed subgraphs if the diameter d is larger than the numerical girth g.

First we recall our notation and terminology.All graphs in this paper are undirected finite simple graphs. Let 0 be a connected graph

with usual distance 0 . We identify 0 with the set of vertices. The diameter of 0, denoted byd0 , is the maximal distance of two vertices in 0. Let u 0. We denote by 0 j (u) the set ofvertices which are at distance j from u.

Let x, y 0 with 0(x, y) = i . DefineC(x, y) := 0i1(x) 01(y),A(x, y) := 0i (x) 01(y)

and B(x, y) := 0i+1(x) 01(y).We say ci exists if ci = |C(x, y)| does not depend on the choice of x and y under the condition0(x, y) = i . Similarly, we say ai exists, or bi exists.

A connected graph 0 with the diameter d0 is said to be distance-regular if ci , ai and bi1exist for all 1 i d0 .

The reader is referred to [1, 2] for more detailed descriptions of distance-regular graphs.Let 0 be a connected graph of the diameter d0 = d 2.For any x, y 0 and 6= 1 0, we define

1 := {z 0 | 0(x, z) 1 for any x 1}and

S(x, y) := {y} C(x, y) A(x, y) = {y} B(x, y).We identify 1 with the induced subgraph on it. A subgraph 1 is called a clique (resp.

coclique) if any two vertices on it are adjacent (resp. non-adjacent).For v 1, 1 is said to be strongly closed with respect to v if S(v, v) 1 for any v 1.

1 is called strongly closed if it is strongly closed with respect to v for all v 1.Singular lines of 0 are the sets of the form {x, y} where (x, y) is an edge in 0. In

particular, a singular line of 0 is always a clique.

01956698/99/080789 + 08 $30.00/0 c 1999 Academic Press

790 A. Hiraki

Let (N P) j be the following condition:(N P) j : If x 0 and L is a singular line with 0(x, L) := min{0(x, z) | z L} = j , then

there is a unique vertex y L such that 0(x, y) = j .We write (N P) 1. Then tr+1 < < tm1 bq . Suppose the following conditions hold.

(i) (SS)

792 A. Hiraki

LEMMA 2.4 [6, Lemmas 2.4 and 2.6]. Let 0 be a distance-regular graph with bq1 > bqand (C R)q holds. Then we have the following.

(1) If (w, x, y, z) is a root of size q, then 9(y, z) 1(w, x).(2) If (SS) 1, bh1 > bh and (SC)h holds. If there exista vertex u and a path (x0, . . . , xh) of length h such that 0(x0, xh) = 0(u, xi ) = h for all0 i h, then ah < ah+1.

REMARK. For the results in this section 0 need not be a distance-regular graph. Suppose0 is a graph such that ci , ai and bi exist for all 0 i q . Then the results are proved by thesame manner as in [6].

Let 1 be a strongly closed subgraph of the diameter q in 0. Then ci and ai of 1 existfor all 1 i q which are the same as those of 0. Moreover, if 1 is a regular graph ofvalency k1, then bi of 1 exists with bi = k1 ci ai for all 0 i q 1, and hence it isdistance-regular.

3. SOME BASIC PROPERTIES

In this section we collect some basic properties and prove the following result.PROPOSITION 3.1. Let 0 be a graph of order (s, t; t2, . . . , tm). Then (SS)

Strongly closed subgraphs in a regular thick near polygon 793

(2) Let (p = pm, pm+1, . . . , ph = p) be a shortest path connecting them. Assume 0(u,p) = h. Then we have 0(u, pi ) = i for all m i h. Since (SS)m holds, we haveS(u, pm) = S(v, pm). This implies pm+1 B(u, pm) = B(v, pm) and 0(u, pm+1) =0(v, pm+1) = m + 1. Inductively, we have pi B(u, pi1) = B(v, pi1) and0(v, pi ) = i for all m + 1 i h. The desired result is proved. 2

Next we show the following well-known result.LEMMA 3.4. Let 2 h d. Suppose a1 and ci exist for all 1 i h. Then the following

conditions are equivalent:(i) (N P)

794 A. Hiraki

From a basic property of graphs we have the following corollary.

COROLLARY 3.5. For a graph of order (s, t; t2, . . . , tm), we have 0 t2 tm .The rest of this section we prove the following result.

LEMMA 3.6. Let q be a positive integer and 0 be a graph of the diameter d0 > q such thatB(x, y) 6= for any x, y 0 with 0(x, y) = i q. Suppose (SC)q holds. Then (SS)qholds.

PROOF. Let (x, y, z) be a triple of vertices with 0(x, y) 1 and 0(x, z) = 0(y, z) =h q . Suppose there exists w S(y, z) S(x, z) to derive a contradiction. Then 0(x, w) =h+1, 0(x, y) = 1 and 0(y, w) = h. Letwh := w and takewi B(x, wi1) B(y, wi1)for h + 1 i q . Then 0(x, wq) = q + 1 and 0(y, wq) = q . Since (SC)q holds, thereexists a strongly closed subgraph 1 of the diameter q containing (y, wq). Then wh 1 as itis on a shortest path between y and wq . Thus z S(y, wh) 1 and x S(z, y) 1. Wehave q + 1 = 0(x, wq) d1 = q , which is a contradiction. The lemma is proved. 2

REMARK. 0 has no induced subgraph K2,1,1 iff (SS)1 holds. More information about therelations among (SS)h, (C R)i and (SC) j , the reader is referred to [6, 7].

4. PROOF OF THE THEOREM

In this section we prove our main theorem. First we prove the following result.

PROPOSITION 4.1. Let 0 be a graph such that ci , ai and bi exist for all i q with a1 > 0and bq1 > bq . Suppose (C R)q and (SS) 0. Then 0(w, v) =0(y, v) = m + 1 from Lemma 3.3(1). Lemma 2.4(2) implies v A(y, p) 1(y, z) andthus v 1(w, x) from the maximality of m. Hence we have p A(w, v) 1(w, x) fromLemma 2.4(2). This is a contradiction. Therefore 1(y, z) 1(w, x).

By symmetry, we have 1(w, x) = 1(y, z). The proposition is proved. 2Next we prove the following result.

PROPOSITION 4.2. Let r and q be positive integers with r + 1 q. Let 0 be a graph withthe numerical girth g = 2r + 2, the diameter d0 q + r such that ci , ai and bi exist for alli q with a1 > 0 and bq1 > bq . Suppose (N P)q and (SS)

Strongly closed subgraphs in a regular thick near polygon 795

LEMMA 4.3. Let 0 be a graph as in Proposition 4.2 satisfying (N P)q and (SS) 1. If th1 < th , then thereexist a vertex u and a path (x0, . . . , xh) of length h in 0 such that 0(x0, xh) = 0(u, xi ) = hfor all 0 i h.

PROOF. Fix a vertex u in 0. First we claim that A(u, w) B(v,w) 6= for any v,w 0h(u) with 0(v,w) = i < h. Suppose A(u, w) B(v,w) = . Then

A(u, w) C(v,w) A(v,w).The right-hand side is a disjoint union of (ti + 1) cliques of size s and the left-hand sidecontains a disjoint union of (th + 1) cliques of size s 1 from Lemma 3.4. This contradictsti th1 < th . Hence our claim is proved.

Take x0 0h(u). Inductively we can take xi A(u, xi1) B(x0, xi1) for all 1 i hfrom our claim. The lemma is proved. 2

PROOF OF THEOREM 1.1. Proposition 3.1 shows that ci , ai1 and bi1 exist for all i m + r such that

ci = ti + 1, ai1 = (ti1 + 1)(s 1) and bi1 = s(t ti1).In particular, (SS)

796 A. Hiraki

From our assumption we have tr < tr+1 and hence br > br+1. Hence (SC)r+1 holds fromProposition 4.2.

Let r+1 h < m. Suppose th1 < th and (SC)h holds. Then ah < ah+1 from Lemmas 4.4and 2.5. Thus th < th+1 and (SC)h+1 holds from Proposition 4.2. Note that ci and ai of astrongly closed subgraph are the same as those of 0. The theorem is proved. 2

REMARK. A regular near 2d-gon of order (s, t; t2, . . . , td) is called a generalized 2d-gonof order (s, t) if t1 = = td1 = 0 and td = t .

Feit and Higman showed that a generalized 2d-gon has d {2, 3, 4, 6}, unless it is anordinary polygon (see [4] or [2, Theorem 6.5.1]).

Let r and m be positive integers with r+1 m. Let 0 be a graph of order (s, t; t2, . . . , tm+r )with s > 1 and 0 = t1 = = tr < tr+1. Theorem 1.1 shows that a graph 0 has a generalized2(r + 1)-gon of order (s, tr+1) as a strongly closed subgraph. Hence we have r {1, 2, 3, 5}from the result of Feit and Higman. This result was first proved in [5].

Here we conjecture the following.CONJECTURE. Let 0 be a regular thick near polygon of the diameter d and the numerical

girth g 6. Then d < g.Suppose 0 is a regular thick near polygon of order (s, t; t2, . . . , td) with the numerical girthg = 2r + 2 6. Suppose 2r + 2 d . Then Corollary 1.2 shows that 0 = t1 = = tr