Distance-regular Subgraphs in a Distance-regular Graph, IV

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1 Europ. J. Combinatorics (1997) 18, Distance-regular Subgraphs in a Distance-regular Graph, IV A KIRA H IRAKI Let be a distance-regular graph with a 1 0, r max j 3 ( c j, a j, b j ) ( c 1, a 1, b 1 ) 2 and a i a 1 c i, for 1 i 2 r. Take any u and in at distance r 1. We show that there exists a collinearity graph of a generalized 2( r 1)-gon of order ( a 1 1, c r 1 1) containing u and as a subgraph in Academic Press Limited 1. I NTRODUCTION Distance-regular graphs have a lot of rich substructures. For example, the Hamming graph H ( d, q ) has H ( h, t ) as its subgraphs for any h d and any t q. The Johnson graph J ( n, d ) and Odd graph O d 1 have similar substructers. It is known that the Petersen graph appears as a subgraph in the M 22 -graph and in the Ivanov Ivanov Faradjev graph. In general, the relation of a distance-regular graph and its substructures represents characteristics of the original graph. For this reason, studying substructures of distance-regular graphs is very important. In fact, this form of study has been carried out by many (see [2], [3, 4. 3], [5] [7] and [11] [15]). To bound the diameter in terms of the valency k is one of the main open problems for distance-regular graphs. The most general known result is the diameter bound of A. A. Ivanov (see [3, 5. 9] and [10]). He shows that the diameter is bounded in terms of the valency k and the numerical girth g. The next stage is to bound the numerical girth g either by an absolute constant or in terms of the valency k. In [12], J. Koolen gives an existence condition for distance-regular subgraphs in a distance-regular graph and obtains an absolute bound on the numerical girth g for a class of distance-regular graphs using a classification of cubic distance-regular graphs. In [5] and [6], the author gives suf ficient conditions on the existence of a Moore graph and a collinearity graph of a Moore geometry in a distance-regular graph with odd numerical girth, and obtain an absolute bound for the numerical girth g for a class of distance-regular graphs using the fact that Moore graphs and collinearity graphs of a Moore geometry that are not ordinary polygons have diameter 2. It is very interesting to find nice subgraphs in a distance-regular graph and to obtain a restriction of parameters of the original graph by using properties of its subgraphs. We believe that a method of this kind is very powerful for the study of distance-regular graphs. For distance-regular graphs of odd numerical girth, this method is used for the case g 5 in [3, 4. 3C] and [7]. We have dealt with the cases g 7 in [5] and [6]. On the other hand, for the cases of the distance-regular graphs of even numerical girth, the case g 4 is treated by Shult Yanushka [13], Brouwer and Wilbrink [3] and Weng [15]. In this paper, we consider the general cases of even numerical girth where g 6 and give a suf ficient condition for the existence of a collinearity graph of a generalized polygon / 97 / $ / 0 ej Academic Press Limited

2 636 A. Hiraki Our main results are Theorem 1. 1 and Corollary In Section 3, we introduce some configurations of vertices and study them. Also, a new condition is defined here. This condition will become a key for constructing distance-regular subgraphs. (In fact, it has been shown that this condition is equivalent for a distance-regular graph to have a distance-regular subgraph : see [8].) In Section 4, we give a suf ficient condition for the existance of a collinearity graph of a generalized polygon in a distance-regular graph. We prove our main results in Section 5. All graphs considered in this paper are undirected finite graphs without loops or multiple edges. Let be a connected graph. We identify with the set of vertices. For two vertices u and in, a walk of length m connecting u and is a sequence of the vertices ( u x 0, x 1,..., x m ) such that each ( x i, x i 1 ) is an edge of. If x j 1 x j 1 for all 1 j m 1, then we say that the walk is a path. We denote by ( u, ) the distance between u and in, i. e. the length of a shortest path connecting u and in. We denote by d : max ( x, y ) 3 x, y that is called the diameter of. Let and j ( u ) : x 3 ( u, x ) j k ( u ) : 1 ( u ). is called a regular graph of alency k if k ( u ) k for all vertices u. For two vertices u and x in with ( u, x ) j, let C ( u, x ) C j ( u, x ) : j 1 ( u ) 1 ( x ), and A ( u, x ) A j ( u, x ) : j ( u ) 1 ( x ) B ( u, x ) B j ( u, x ) : j 1 ( u ) 1 ( x ). is said to be distance - regular if c j ( ) : C j ( u, x ), a j ( ) : A j ( u, x ) and b j ( ) : B j ( u, x ) depend only on j ( u, x ) rather than individual vertices. It is clear that is a regular graph of valency k b 0 ( ) if is distance-regular. Sometimes we omit the suf fix when the concerning graph is clear. The numbers c j, a j and b j are called the intersection numbers of. Let r ( ) : max j 3 ( c j, a j, b j ) ( c 1, a 1, b 1 ). A circuit of length m is a sequence of distinct vertices ( x 0, x 1,..., x m 1 ) such that ( x i, x i 1 ) is an edge of for all 0 i m 1, where x m x 0. Let be a distance-regular graph. We denote by g the numerical girth of, that is the length of a shortest circuit which does not contain triangles. It is well known that where r r ( ). g 2 r 2 if c r 1 2, 2 r 3 if c r 1 1,

3 Distance - regular subgraphs, IV 637 We say that a minimal circuit for a circuit of length g is one that does not contain triangles. The reader is referred to [1, 2] for a general theory of distance-regular graphs. Let be a graph, x, y and Z. We define by x : x 1 ( x ), Z : " z, z Z ( x, Z ) : min ( x, z ) 3 z Z, P ( x, y ) : z 3 ( x, z ) ( z, y ) ( x, y ), and P ( x, Z ):! P ( x, z ) z Z S ( x, y ) : y C ( x, y ) A ( x, y ) y B ( x, y ). A subset of vertices is called a clique (respectively coclique ) if any two of vertices in it are adjacent (respectively non-adjacent). Singular lines of are the sets of the form x, y, where ( x, y ) is an edge in. It is well known that there is no induced subgraph K 2, 1, 1 in if f x, y x, y A 1 ( x, y ) for any edge ( x, y ). In particular, a singular line of is always a clique. Let ( NP ) j be the following condition : ( NP ) j : If x and L is a singular line with ( x, L ) j, then there is a unique ertex y L such that ( x, y ) j. A distance-regular graph of diameter d 2 is called a regular near polygon if ( NP ) j holds for all 1 j d 1. Moreover, if a 1 0, the regular near polygon is called thick. A collinearity graph of a generalized 2 d - gon of order ( s, t ) is an important example of regular near polygons which has diameter d, valency s ( t 1), c d t 1 and c i 1, for all 1 i d 1. More detailed descriptions of a regular near polygon and a generalized polygon will be found in [1, III. 3] and [3, ]. In this paper, we prove the following results. T HEOREM Let be a distance - regular graph with a 1 s 1 0, r ( ) r 2 and c r 1 t 1. Suppose that a i a 1 c i for all 1 i 2 r. Take any u and in at distance r 1. Then there exists a collinearity graph of a generalized 2( r 1)- gon ( u, ) of order ( s, t ) containing u and as a subgraph in. In particular, ( x, y ) ( u, ) for any x and y in ( u, ) at distance r 1. C OROLLARY Let be a thick regular near polygon with diameter d and r r ( ). Suppose that 2 r 1 d. Then r 1, 2, 3, 5 and 2 c r 1 1 c r P RELIMINARIES In this section, we first collect some basic properties for a distance regular-graph of diameter d and r ( ) r 2. Let h, i and s be positive integers. L EMMA If 2 h d 1, then the following conditions are equi alent : (i) there exists no induced subgraph K 2, 1, 1 and a i a 1 c i for all 1 i h ; (ii) ( NP ) i holds for all 1 i h ;

4 638 A. Hiraki (iii) for any 2 i h and any pair of ertices u and x at distance i, we ha e A i ( u, x )! A 1 ( z, x ). ( ) z C i ( u,x) P ROOF. (i) ï (ii) Note that there is no induced subgraph K 2, 1, 1 if f ( NP ) 1 holds. We have our assertion from the proof of [3, Theorem ]. (ii) é (iii) Let y A i ( u, x ). Then there exists z x, y i 1 ( u ) as ( NP ) i holds. Hence y is in the right-hand side of ( ). Conversely, take any y from the right-hand side of ( ) ; then y A i ( u, x ) because ( NP ) i 1 holds. (iii) é (i) has no induced subgraph K 2, 1, 1 if f ( ) holds for i 2. For all 2 i h, the right-hand side of ( ) is a disjoint union. Thus we have a i A i ( u, x )! A 1 ( z, x ) a 1 c i. z C i ( u,x) In this paper, we identify a subset of vertices X with the induced subgraph on it. We also write X 1 ( x ) 1 ( x ) X. D EFINITION Let X, Z. A non-empty set X is called a (, s )- subgraph with respect to Z if the following conditions hold : (i) X 1 ( x ) is not a clique for any x X ; (ii) ( x, z ) s for any x X and z Z. L EMMA Let w, x and X be a (, r 1)- subgraph with respect to w. If x i ( w ) X for some 1 i r, then B i ( w, x ) X. P ROOF. Suppose that B i ( w, x ) X. Then we have X 1 ( x ) C i ( w, x ) A i ( w, x ). The left-hand side is not a clique, from our definition. However, the right-hand side is a clique, as 1 i r. This is a contradiction. L EMMA Let ( x, y, z ) be a triple of ertices in with ( x, z ) ( y, z ) i and ( x, y ) 1. Suppose that ( NP ) i holds. Then B i ( x, z ) B i ( y, z ). P ROOF. Suppose that there exists w B i ( x, z ) B i ( y, z ). Then we have ( x, w ) i 1 and hence ( y, w ) ( y, z ) i. As ( NP ) i holds, there exists z, w i 1 ( y ). Since ( x, z ) i and ( x, w ) i 1, we have ( x, ) i 1 as ( NP ) i holds. We have i 1 ( x, ) ( x, y ) ( y, ) i. This is a contradiction. Comparing the sizes of both sides, we have the assertion. 3. A C ONFIGURATION OF V ERTICES, A C ONRON Let be a distance-regular graph with a 1 0, r ( ) r 2, and diameter d 2 r 1. All singular lines of have size a 1 2 as c 2 1. Let us consider the following condition : ( BB ) : For any i 2 r and any triple of ertices ( x, y, z ) in with ( x, z ) ( y, z ) i and ( x, y ) 1, B i ( x, z ) B i ( y, z ) holds. We remark that a distance-regular graph as in Theorem 1. 1 satisfies the condition

5 Distance - regular subgraphs, IV 639 ( NP ) i for all 1 i 2 r from Lemma 2. 1, and hence it satisfies ( BB ) from Lemma For the rest of this section, we assume that satisfies ( NP ) r 1 and ( BB ). L EMMA C ( x, y ) is a coclique for any x, y with ( x, y ) 2 r 1. P ROOF. We prove the assertion by induction on i ( x, y ). For the cases i r, the assertion is clear, since c i 1. Let r 1 i 2 r 1. Suppose that there exists an edge ( w, z ) in C i ( x, y ), to derive a contradiction. Let C i 1 ( w, x ) C i ( y, x ). Then (, z ) i 2, i 1. If (, z ) i 1, then y B i 1 ( x, z ) B i 1 (, z ), and thus ( x,, z ) violates the condition ( BB ). If (, z ) i 2, then there exists an edge ( w, z ) in C i 1 (, y ), which contradicts the inductive assumption. The lemma is proved. L EMMA Let x and ( 0, 1,..., r, r,..., 1, 0 ) be a minimal circuit. If ( x, 0 ) ( x, 0 ) r 1 and j B r j ( x, j 1 ) for all 1 j r, then j B r j ( x, j 1 ) for all 1 j r. P ROOF. Note that ( NP ) r 1 holds. Since ( x, 0 ) ( x, 0 ) r 1, there exists z A 1 ( 0, 0 ) r ( x ). Then we have ( z, r ) ( z, r ) r 1 and hence there exists z A 1 ( r, r ) r ( z ). As 2 r ( x, r ) ( r, z ) ( x, z ) ( x, z ) ( z, z ) 2 r, we have z 2 r ( x ). Since C ( x, r ) is a coclique, we obtain r 2 r 1 ( x ) and thus j r j 1 ( x ) for all 0 j r. This is the desired result. D EFINITION Let w, x, y, z. Then : (1) A quadruple of vertices ( w, x, y, z ) is called a root if ( w, x ) ( y, z ) r 1, y S ( x, w ) and z S ( w, x ). (2) ( x, y, z ) is called a conron if there exist three sequences of vertices ( x 0, x 1,..., x m x ), ( y 0, y 1,..., y m y ) and ( z 0, z 1,..., z m z ) such that ( x 0, z 0, y 0, z 0 ), ( x i 1, z i 1, x i, z i ) and ( y i 1, z i 1, y i, z i ) are roots for all 1 i m. In particular, we call these sequences of vertices a c - sequences. (3) The condition ( CR ) is defined as follows : ( CR ) : S ( x, z ) S ( y, z ) for any conron ( x, y, z ). In the above definition, the vertices need not be dif ferent. L EMMA Let x, x and ( 0, 1,..., r, r,..., 1, 0 ) be a minimal circuit such that ( x, 0, x, 0 ) is a root. If j B r j ( x, j 1 ) for all 1 j r, then j B r j ( x, j 1 ) for all 1 j r. P ROOF. If ( x, 0 ) r 1, then we have j B r j ( x, j 1 ) B r j ( x, j 1 ) for all 1 j r from the condition ( BB ). The assertion follows from Lemma If ( x, 0 ) r 1, then we also obtain j B r j ( x, j 1 ) B r j ( x, j 1 ) for all 1 j r from Lemma 3. 2 and the condition ( BB ). Hence we may assume that ( x, 0 ) ( x, 0 ) r. Let x A 1 ( x, x ). Then we have ( x, 0 ) ( x, 0 ) r 1 as ( NP ) r 1 holds. Thus we obtain j B r j ( x, j 1 )

6 640 A. Hiraki for all 1 j r, as in the first case. Since ( BB ) holds, we have j B r j ( x, j 1 ) B r j ( x, j 1 ) for all 1 j r. This proves the lemma. L EMMA Let x, z, z and X be a (, r 1)- subgraph with respect to z, z. If ( z, z ) 1 and x r 1 ( z ) X, then there exists x X such that ( z, x, z, x ) is a root. P ROOF. Since x r 1 ( z ) X, we have ( x, z ) r, r 1. If ( x, z ) r 1, then we can take x for x. If ( x, z ) r, then there exists x B r ( z, x ) X, from Lemma In either case, we have the assertion. L EMMA Let ( x, y, z ) be a conron and let ( z, 0, 1,..., r ) be a path of length r. If i B ( x, i 1 ) for all 1 i r, then i B ( y, i 1 ) for all 1 i r. P ROOF. We prove by induction on the length m of c-sequences of the conron. If m 0, then the assertion follows from the condition ( BB ). Let ( x, y, z ) be a conron with a c-sequences ( x 0, x 1,..., x m 1 x ), ( y 0, y 1,..., y m 1 y ) and ( z 0, z 1,..., z m 1 z ). Then we have that ( x m, y m, z m ) is a conron with a c-sequences If z m z, then ( x 0, x 1,..., x m ), ( y 0, y 1,..., y m ) and ( z 0, z 1,..., z m ). B ( x, i ) B ( x m, i ) B ( y m, i ) B ( y, i ) for all i from the condition ( BB ) and the inductive assumption. If z m z, then ( z m, r ) r 1 as, otherwise, z m S ( r, z ) S ( 1, z ) and ( z, z m ) is an edge in C r 2 ( x, 1 ), contradicting Lemma Take u r C ( z m, r ) r 1 and let P ( z m, u r ) ( u 0, u 1,..., u r ). Then ( 0, 1,..., r, u r,..., u 1, u 0 ) is a minimal circuit. As ( x, z, x m, z m ) is a root, we have u i B ( x m, u i 1 ) for all 1 i r, from Lemma From the inductive assumption, u i B ( y m, u i 1 ) for all 1 i r. As ( y m, z m, y, z ) is a root, we obtain i B ( y, i 1 ) for all 1 i r from Lemma The lemma is proved. P ROPOSITION Under the hypothesis ( BB ), the condition ( CR ) holds. P ROOF. Let ( x, y, z ) be a conron. Let i B ( x, i 1 ) for all 1 i r, where 0 z. Then we have i B ( y, i 1 ) for all 1 i r from the previous lemma. In particular, we have B ( x, z ) B ( y, z ) and thus S ( x, z ) S ( y, z ). This is the desired result. 4. D ISTANCE-REGULAR S UBGRAPHS Our purpose in this section is to show the following result : P ROPOSITION Let be a distance - regular graph with a 1 0, r ( ) r 2 and c r 1 c 2. Suppose that ( NP ) r 1 and ( CR ) hold. Then for any u and in at distance r 1, there exists a collinearity graph of a generalized 2( r 1)- gon ( u, ) of order ( a 1 1, c 1) containing u and as a subgraph in. In the following, let be a distance-regular graph as in Proposition From Proposition in [3], we have b r b r 1.

7 Distance - regular subgraphs, IV 641 L EMMA Let ( x, y, z ) be a conron. If x, z such that ( x, z, x, z ) is a root, then there exists y S ( z, y ) such that ( y, z, y, z ) is a root. In particular, ( x, y, z ) is a conron. P ROOF. Since ( CR ) holds, we have z S ( x, z ) S ( y, z ). This implies that ( y, z ) r, r 1. If ( y, z ) r 1, then we set y y. If ( y, z ) r, then there exists y B r ( z, y ) B r 1 ( z, y ). In either case, we have y S ( z, y ) such that ( y, z, y, z ) is a root. The rest of proof is clear. L EMMA For any x, y r 1 ( z ), define x y if f ( x, y, z ) is a conron. Then this is an equi alence relation on r 1 ( z ). P ROOF. Reflexivity and symmetricity are clear. We show transitivity. Let ( x, y, z ) and ( y, w, z ) be conrons with c-sequences and ( x 0, x 1,..., x m ), ( y 0, y 1,..., y m ), ( z 0, z 1,..., z m ) ( y 0, y 1,..., y s ), ( w 0, w 1,..., w s ), ( z 0, z 1,..., z s ). From Lemma 4. 2, there exists a sequence of vertices ( x 0, x 1,..., x s x ) such that ( x i 1, z i 1 x i, z i ) is a root for all 0 i s. Then ( x, w, z ) is a conron, with a c-sequences and ( x 0, x 1,..., x m, x s, x s 1,..., x 0, x 0, x 1,..., x s ), ( y 0, y 1,..., y m, y s, y s 1,..., y 0, w 0, w 1,..., w s ) ( z 0, z 1,..., z m, z s, z s 1,..., z 0, z 0, z 1,..., z s ). The lemma is proved. Take any u, at distance r 1. Define an equivalence relation on r 1 ( u ) as in the previous lemma. Let be the equivalence class containing and ( u, ) : P ( u, ). It is clear that 1 ( x ) is not a clique for all x. L EMMA Let z. If is a (, r 1)- subgraph with respect to P ( u, z ), then ( x, y, z ) is a conron for any x, y r 1 ( z ). P ROOF. We prove by induction on h ( u, z ). The case h 0 is clear. Let 1 h and z C ( u, z ). For any x, y r 1 ( z ), there exist x, y such that ( z, x, z, x ) and ( z, y, z, y ) are roots from Lemma As P ( u, z ) P ( u, z ), we have that ( x, y, z ) is a conron, from the inductive assumption. So ( x, y, z ) is a conron. L EMMA Let w. Then j ( w ), for any j r 2. P ROOF. We prove our assertion by induction on h ( u, w ). The assertion is clear for the case h 0. Let 1 h and z C ( u, w ). From the inductive assumption, is a (, r 1)-subgraph with respect to P ( u, z ). From Lemma 2. 3, we can take w i B ( z, w i 1 ) for all 2 i r 1, where w 1 w. So we have x : w r 1 r 1 ( z ) such that w C r 1 ( x, z ).

8 642 A. Hiraki Suppose that there exists y j ( w ) for some j r 2, to derive a contradiction. Since r 2 ( w, y ) ( w, z ) ( z, y ) 1 ( r 1) r 2, we have y r 1 ( z ) r 2 ( w ). This implies that y, x r 1 ( z ) such that w S ( x, z ) S ( y, z ), which contradicts Lemma 4. 4 as ( CR ) holds. L EMMA Let x u and x C ( u, x ). Then : (1) x A 1 ( x, x ) C ( u, x ) A ( u, x ) ; (2) there exists z r 1 ( x ) such that C r 1 ( z, x ) A r 1 ( z, x ) ; (3) C (, u ) A (, u ). P ROOF. Note that ( NP ) r 1 holds. If ( u, x ) r 1, then x A 1 ( x, x ) C r 1 ( u, x ) A r 1 ( u, x ) P ( u, x ). Thus we may assume that ( u, x ) h r. Then x A 1 ( x, x ) C ( u, x ) A ( u, x ). Since x, there exists u such that x P ( u, u ). Let z C ( u, u ) C ( x, u ), z h ( u ) P ( u, z ), P ( x, u ) ( x 0, x 1,..., x m ) and P ( z, u ) ( z 0, z 1,..., z m ), where m r 1 h. Then ( x, z ) r 1, ( x, z ) r and ( x i 1, z i 1, x i, z i ) is a root for all 1 i m. Take any y A 1 ( x, x ). Then ( y, z ) r 1 as ( NP ) r 1 holds. Hence we have that ( x, y, z ) is a conron and thus there exists a sequence of vertices ( y y 0, y 1,..., y m ) such that ( y i 1, z i 1, y i, z i ) is a root and ( x i, y i, z i ) is a conron for all 1 i m from Lemma In particular, y m as ( x m, y m, z m u ) is a conron. So y P ( u, y m ). Hence we have x A 1 ( x, x ), and (1) is proved. Let w C r 1 ( z, x ) x and P ( z, w ) ( w 0, w 1,..., w r ). Then ( u, w h, u ) is a conron with a trinetra ( x 0, x 1,..., x m ), ( x, w r,..., w h ) and ( z 0, z 1,..., z m ). Thus w h and w P ( u, w h ) ; whence we have C r 1 ( z, x ). Note that A 1 ( x, w ) for any w C r 1 ( z, x ) from (1). We have A r 1 ( z, x )! A 1 ( x, w ), w C ( z,x) by Lemma This proves (2). Since C r 1 (, u ) P ( u, ), (3) is proved as above. L EMMA is a regular graph of alency k c ( a 1 1). P ROOF. Let x. From Lemma 4. 6(2), (3), there exists z r 1 ( x ) such that C r 1 ( z, x ) A r 1 ( z, x ). Also, we have B r 1 ( z, x ) from Lemma Thus 1 ( x ) C r 1 ( z, x ) A r 1 ( z, x ) c ca 1. The lemma is proved. L EMMA Let z. Then C ( z, x ) A ( z, x ), for any x z. P ROOF. We prove the assertion by induction on h ( u, z ). The case h 0 is proved in Lemma 4. 6(1). Assume that 1 h and let z C h ( u, z ). Take any x z. If ( z, x ) r, then we have C ( z, x ) A ( z, x ) C ( z, x ) A ( z, x ),

9 Distance - regular subgraphs, IV 643 from a basic property of distance-regular graphs and our inductive assumption. Hence we may assume that ( z, x ) r 1 and B ( z, x ) from Lemma Thus we have 1 ( x ) C ( z, x ) A ( z, x ). Comparing the sizes of both sides using Lemma 4. 7, we have 1 ( x ) C ( z, x ) A ( z, x ). The assertion follows. R EMARK. A subgraph satisfying the condition of Lemma 4. 8 is called a strongly closed subgraph in [14] or a weak - geodetically closed subgraph in [15]. P ROOF OF P ROPOSITION Take any z, x. From Lemma 4. 8, P ( z, x ). Thus ( z, x ) ( z, x ). From Lemmas 4. 5 and 4. 7, is a regular graph of valency k c ( a 1 1) and diameter d r 1. Lemma 4. 8 implies that c i ( ) c i ( ), a i ( ) a i ( ) and b i ( ) k c i ( ) a i ( ), for all 1 i d r 1. This implies that the graph is a collinearity graph of a generalized 2( r 1)-gon of order ( a 1 1, c 1). This completes the proof of Proposition P ROOF OF THE T HEOREM Our purpose in this section is to prove Theorem 1. 1 and Corollary First, we show the following results. P ROPOSITION Let be a distance - regular graph as in Proposition 4. 1 and let u,, x, y. Then : (1) if ( u,, x, y ) is a root, then ( x, y ) ( u, ) ; (2) if ( x, y ) ( u, ) r 1 and x, y ( u, ), then ( x, y ) ( u, ) ; (3) if ( NP ) r 2 holds, then 2 c r 1 1 c r 2. P ROOF. (1) Let ( u, ) and ( x, y ). From Lemma 4. 8, we have u,. (,, u ) is a conron for any. So there exists y S ( u, ) such that (, u, y, x ) is a root and ( y, y, x ) is a conron by Lemma Then y and hence S ( x, y ). This implies that. Also, from Lemma 4. 8, P ( u, ). As and are distance-regular graphs with the same parameter, we have. (2) We prove the assertion by induction on h ( u, x ). The case h 0 is clear. So we may assume that h 1, and that x C ( u, x ). Then there exists y such that ( x, y, x, y ) is a root from Lemma Thus we have from (1) and our inductive hypothesis. (3) Suppose that c r 2 2 c r 1 2. Then ( x, y ) ( x, y ) ( u, ) a r 2 a 1 c r 2 a 1 (2 c r 1 2) 2 a r 1 2 a 1. Let u, x with ( u, x ) r 2. Take C r 2 ( x, u ). We have (, ) 2. Let (, x ) and (, x ). We remark that A r 1 (, x ) A r 1 (, x ) A r 2 ( u, x ) as ( NP ) r 1 holds. Thus we obtain A r 1 (, x ) A r 1 (, x ) A r 1 (, x ) A r 1 (, x ) A r 1 (, x ) A r 1 (, x ) 2 a r 1 a r 2 2 a 1.

10 644 A. Hiraki So we have x 1 A r 1 (, x ) A r 1 (, x ) and x 2 A r 1 (, x 1 ) A r 1 (, x 1 ) A 1 ( x, x 1 ). From Lemma 4. 8, we have x 1, x 2. Inductively, we can take x i 1 A r 1 (, x i ) A r 1 (, x i ) A 1 ( x, x i ), for all 2 i r. Then x, x r 1 such that ( x, x r 1 ) r 1. This implies that ( x, x r 1 ) from (2). As,, we have u C 2 (, ). This contradicts ( x, u ) r 2 d. P ROOF OF T HEOREM Since r 2, c 2 1 and has no induced subgraph K 2, 1, 1. satisfies ( NP ) i for all 1 i 2 r from Lemma Lemma 2. 4 implies that satisfies the condition ( BB ), whence ( NP ) r 1 and ( CR ) hold from Proposition Our theorem is a direct consequence of Proposition 4. 1 and Lemma 5. 1(2). To prove Corollary 1. 2, we use the following famous theorem of Feit and Higman. T HEOREM 5. 2 ([4], [3, Theorem ]). A collinearity graph of a generalized 2( r 1)- gon has r 1, 2, 3, 5, unless it is an ordinary polygon. P ROOF OF C OROLLARY From Theorem 1. 1, there exists a collinearity graph of a generalized 2( r 1)-gon of order ( s, t ) as a subgraph in. Thus Theorem 5. 2 implies that r 1, 2, 3, 5. The second part is a direct consequence of [2, Theorem 6] and Proposition 5. 1(3). Thus the corollary is proved. R EMARK. The subgraph chasing technique that we use in [5 7] and in this paper is applicable to many classes of distance-regular graphs and gives us a lot of good results. In fact, the author succeeded in proving that the condition ( CR ) is an equivalent condition for a distance-regular graph to have a strongly closed subgraph of diameter r 1, where r r ( ). We have found satisfying several kinds of distance-regular graphs satisfying the condition ( CR ). We will introduce them in the succeeding papers [8] and [9]. We strongly believe that a method of this kind is very powerful for the study of the substructure of a distance-regular graph. A CKNOWLEDGEMENT The author was supported in part by a grant from the Japan Society for Promotion of Sciences. R EFERENCES 1. E. Bannai and T. Ito, Algebraic Combinatorics I, Benjamin-Cummings, Menlo Park, California, A. E. Brouwer and H. A. Wilbrink. The structure of near polygons with quads, Geom. Ded. 14 (1983), A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance - regular Graphs, Springer-Verlag, Berlin, W. Feit and G. Higman, The non-existence of certain generalized polygons, J. Algebra, 1 (1964), A. Hiraki, Distance-regular subgraphs in a distance-regular graph, I, Europ. J. Combin., 16 (1995), A. Hiraki, Distance-regular subgraphs in a distance-regular graph, II, Europ. J. Combin. 16 (1995), A. Hiraki, Distance-regular subgraphs in a distance-regular graph, III, Europ. J. Combin., 17 (1996), A. Hiraki, Distance-regular subgraphs in a distance-regular graph, V, to appear in Europ. J. Combin. 9. A. Hiraki, Distance-regular subgraphs in a distance-regular graph, VII, preprint.

11 Distance - regular subgraphs, IV A. A. Ivanov, Bounding the diameter of a distance-regular graph, So. Math. Dokl., 28 (1983), J. Koolen, A new condition for distance-regular graphs, Europ. J. Combin., 13 (1992), J. Koolen, On subgraphs in distance-regular graphs, J. Algebraic Combin., 1 (1992), E. Shult and A. Yanushka, Near n -gons and line systems, Geom. Ded., 9 (1980), H. Suzuki, On strongly cloaed subgraphs of highly regular graphs, Europ. J. Combin., 16 (1995), C-W. Weng, Weak-geodetically closed subgraphs in distance-regular graphs, preprint. Recei ed 1 5 January and accepted in re ised form 4 No ember A KIRA H IRAKI Di ision of Mathematical Sciences, Osaka Kyoiku Uni ersity Kashiware, Osaka 582, Japan