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Europ . J . Combinatorics (1997) 18 , 65 – 74
On the Characterization of the Folded Johnson Graphs and the Folded Halved Cubes by their Intersection Arrays
K LAUS M ETSCH
We show that the following distance-regular graphs are uniquely determined by their intersection arrays : the folded Johnson Graphs of diameter d у 8 and the folded halved cubes of diameter d у 8 . ÷ 1997 Academic Press Limited
1 . I NTRODUCTION This paper is a contribution to the characterization problem of distance-regular graphs by their intersection arrays . This problem has received much attraction in the past and many results have been obtained . For detailed information , we refer the reader to Brouwer , Cohen and Neumaier [3] . All graphs are assumed to be finite , undirected and without loops or multiple edges .
For a vertex x of a graph ⌫ , we denote by ⌫ i ( x ) , i у 0 , the set consisting of the vertices at distance i from x . The graph ⌫ is called distance - regular with diameter d and intersection array
͕ b 0 , b 1 , . . . , b d Ϫ 1 ; c 1 , c 2 , . . . , c d ͖ , (1)
if d is the maximum distance between two vertices of ⌫ , and if ͉ ⌫ 1 ( x ) ʝ ⌫ i ϩ 1 ( y ) ͉ ϭ b i and ͉ ⌫ 1 ( x ) ʝ ⌫ i Ϫ 1 ( y ) ͉ ϭ c i for any two vertices x and y at distance i . The folded Johnson graph J (2 m , m ) , m у 3 an integer , is a distance-regular graph with diameter d ϭ m / 2 and intersection array (1) satisfying 2 2 b i ϭ ( m Ϫ i ) , c i ϭ i (0 р i р d Ϫ 1) , d 2 , if m is odd ; (2) c ϭ d ͭ 2 d 2 , if m is even . It can be defined as follows . The vertices are the partitions of a 2 m -set X into two m -sets , two partitions being adjacent if their common refinement is a partition of X into sets of size 1 , 1 , m Ϫ 1 , m Ϫ 1 . The folded hal ed 2 m - cube , m у 3 an integer , is a distance-regular graph with diameter d : ϭ m / 2 and intersection array (1) in which
b i ϭ ( m Ϫ i )(2 m Ϫ 2 i Ϫ 1) , c i ϭ i (2 i Ϫ 1) (0 р i р d Ϫ 1) , d (2 d Ϫ 1) , if m is odd ; (3) c ϭ d ͭ 2 d (2 d Ϫ 1) , if m is even . It can be defined as follows . The vertices are the partitions of a 2 m -set into two subsets of even size , two partitions ͕ A , A Ј ͖ and ͕ B , B Ј ͖ being adjacent if min ( ͉ A ⌬ B ͉ , ͉ A ⌬ B Ј ͉ ) ϭ 2 (where ⌬ denotes the symmetric dif ference) . Bussemaker and Neumaier [4] proved that the folded Johnson graphs and the folded halved cubes of diameter d у 154 are characterized by their intersection array . Their proof goes as follows . Given a graph ⌫ with the same intersection array as a folded Johnson graph or a folded halved cube , they used the theory of association schemes (see Bannai and Ito [1]) and a result of Terwilliger [7] to obtain Ϫ 2 Ϫ 2 / ( m Ϫ 3) as a 65 0195-6698 / 97 / 010065 ϩ 10 $25 . 00 / 0 ej950082 ÷ 1997 Academic Press Limited 66 K . Metsch lower bound for the smallest eigenvalue Θ of the neighbourhood of a vertex . Using their main result , that is the set consisting of the smallest eigenvalues of all finite graphs has a gap at Ϫ 2 , they concluded that Θ у Ϫ 2 . Then , by a well-known theorem of Cameron , Goethals , Seidel and Shult [5] , it follows that each neighbourhood has to be a line graph , and this information is suf ficient to determine the graph . It is the purpose of this paper to improve the bound on d . In our situation the elegant method described above cannot be applied , because the bound for Θ is too weak . However , it is still strong enough to show that certain small graphs cannot be induced subgraphs in the neighbourhood of a vertex . This information is suf ficient to obtain the same conclusion ; namely , that the neighbourhood of each vertex is a line graph , and the proof can be completed .
T HEOREM 1 . 1 . The folded Johson graphs of diameter d у 8 are uniquely determined by their intersection array .
T HEOREM 1 . 2 . The folded hal ed cubes of diameter d у 8 are uniquely determined by their intersection array .
We remark that these theorems , together with Theorem 3 . 3 in [4] , classify all pseudo-partition graphs of diameter d у 8 .
2 . T HE C HARACTERIZATION Throughout , let m у 16 be an integer , d : ϭ m / 2 , and ⌫ a distance-regular graph of diameter d the parameters of which satisfy either (2) or (3) . By k : ϭ b 0 we denote the valency , by : ϭ k Ϫ b 1 Ϫ c 1 the number of vertices adjacent to two adjacent vertices , and by : ϭ c 2 the number of vertices adjacent to two non-adjacent vertices . For each vertex x we denote by ⌫ ( x ) the graph induced by ⌫ on ⌫ 1 ( x ) . For two vertices x and y at distance one or two , we put ⌬ ( x , z ) : ϭ ⌫ 1 ( x ) ʝ ⌫ 1 ( y ) . Two vertices are called independent if they are not adjacent . An anticlique is a set of mutually independent vertices . A clique is a set of mutually adjacent vertices . An k - claw , k у 2 , consists of k independent vertices u 1 , . . . , u k and a vertex adjacent to all of them (the 5-claw is shown in Figure 1 below) . If ϭ 4—that is , if ⌫ has the parameters of a folded Johnson graph—then ϭ 2( m Ϫ 1) and k ϭ m 2 . If ϭ 6—that is , if ⌫ has the parameters of a folded halved cube—then ϭ 4( m Ϫ 1) and k ϭ m (2 m Ϫ 1) . We are interested in the smallest eigenvalue of the adjacency matrix of the graphs ⌫ ( x ) . Using a result of Terwilliger [7] , the following was proved by Bussemaker and Neumaier ([4] , Step 6 in the proof of Theorem 3 . 3) .
R ESULT 2 . 1 . For e ery ertex x , the eigen alues Θ of the adjacency matrix of ⌫ ( x ) satisfy Θ у Ϫ 2 Ϫ 2 / ( m Ϫ 3) .
Given an induced subgraph ⌫ Ј of ⌫ ( x ) , the adjacency matrix A Ј of ⌫ Ј is a principal
–2·236068 Ϫ 2 . 236068 F IGURE 1 . The 5-claw and its minimum eigenvalue . Johnson graphs and hal ed cubes 67 submatrix of the adjacency matrix A of ⌫ ( x ) . It is well-known that , since A Ј and A are symmetric , the smallest eigenvalue of A Ј is not smaller than the smallest eigenvalue of A . This will give strong restrictions .
C OROLLARY 2 . 2 . For e ery ertex x , the eigen alues Θ of the induced subgraphs of Θ – 2 ⌫ ( x ) satisfy у Ϫ 2 Ϫ 13 Ͼ Ϫ 2 . 153846 .
This local information is enough to determine the structure of ⌫ ( x ) . For example , if x is a vertex , then the graph in Figure 1 cannot be an induced subgraph of ⌫ ( x ) .
R ESULT 2 . 3 . Suppose that Q ϭ x 1 y 1 x 2 y 2 is a quadrangle ; that is , d ( x 1 , x 2 ) ϭ d ( y 1 , y 2 ) ϭ 2 and d ( x i , y j ) ϭ 1 for i ,j ϭ 1 , 2 . Then d ( x 1 , u ) ϩ d ( x 2 , u ) ϭ d ( y 1 , u ) ϩ d ( y 2 , u ) for e ery ertex u at distance one from Q .
P ROOF . The parameters of ⌫ satisfy c 2 Ϫ b 2 ϭ c 1 Ϫ b 1 ϩ a 1 ϩ 2 . A result of Terwilli- ger [6] (see Theorem 5 . 2 . 1 of [3] for a proof) says that the assertion of the lemma holds for every distance-regular graph of diameter at least three satisfying this equation . ᮀ
L EMMA 2 . 4 . Suppose that x is adjacent to non - adjacent ertices y 1 and y 2 . Then there exists at most one quadrangle containing y 1 , x and y 2 ; that is , y 1 and y 2 ha e Ϫ 2 or Ϫ 1 common neighbours in ⌫ ( x ) .
P ROOF . This is an immediate consequence of Result 2 . 3 : if y 1 and y 2 have a common neighbour x Ј ϶ x that is not adjacent to x , then xy 1 x Ј y 2 is a quadrangle , so every common neighbour z ϶ x , x Ј of y 1 and y 2 satisfies d ( x , z ) ϩ d ( x Ј , z ) ϭ d ( y 1 , z ) ϩ d ( y 2 , z ) ϭ 2 , which implies that x ϳ z . ᮀ
L EMMA 2 . 5 . Suppose that x ϳ y and that u 1 , u 2 and u 3 are three independent ertices of ⌬ ( x , y ) . Then ⌬ ( u i , u j ) ‘ ⌬ ( x , y ) ʜ ͕ x , y ͖ for 1 р i Ͻ j р 3 .
P ROOF . Consider a vertex z ⌬ ( u i , u j ) , with z ϶ x , y . It follows from Lemma 2 . 4 that x ϳ z or y ϳ z . W . l . o . g ., x ϳ z . Assume that z ϳ ͉ y . Then yu 1 zu 2 is a quadrangle with d ( u 1 , u 3 ) ϩ d ( u 2 , u 3 ) ϭ 4 and d ( y , u 3 ) ϩ d ( z , u 3 ) р 1 ϩ 2 ϭ 3 , since x ϳ u 1 , u 2 , u 3 , z . This contradicts Result 2 . 3 . ᮀ
L EMMA 2 . 6 . If x ⌫ , then ⌫ ( x ) does not contain a 4- claw .
P ROOF . Assume that there exists an independent set U ϭ ͕ u 1 , u 2 , u 3 , u 4 ͖ ‘ ⌬ ( x , y ) for some vertex y ⌫ 1 ( x ) . Let f i be the number of vertices in ⌬ ( x , y ) that are adjacent to exactly i of the vertices u 1 , u 2 , u 3 and u 4 , i ϭ 0 , 1 , 2 , 3 , 4 . Since ⌫ ( x ) does not contain a 5-claw (see Corollary 2 . 2 and Figure 1) , we have f 0 ϭ ͉ ͕ u 1 , u 2 , u 3 , u 4 ͖͉ ϭ 4 . Let r i be the number of vertices in ⌬ ( x , y ) that are adjacent to u i , i ϭ 1 , 2 , 3 , 4 . Then
4 4 4 f i ϭ ͉ ⌬ ( x , y ) ͉ ϭ , f i i ϭ r 1 ϩ r 2 ϩ r 3 ϩ r 4 , and f i i ( i Ϫ 1) ϭ 12( Ϫ 2) . i ϭ 0 i ϭ 0 i ϭ 0 The last equation holds because , for i ϶ j , by Lemma 2 . 5 exactly Ϫ 2 common neighbours of u i and u j lie in ⌬ ( x , y ) . It follows that
4 4 4 1– r 1 ϩ r 2 ϩ r 3 ϩ r 4 ϭ f i i р f i ϩ 2 f i i ( i Ϫ 1) Ϫ f 0 ϭ ϩ 6 Ϫ 16 . i ϭ 0 i ϭ 0 i ϭ 1 68 K . Metsch
u3 u3 vi
y y u3 u2 u3 u2 14
u v 1 j u1
Ϫ–2·175327 2 . 175327 Ϫ–2·176194 2 . 176194 F IGURE 2 . Two graphs and their smallest eigenvalues .
1– We may assume that r 1 у r 2 у r 3 у r 4 . Then r 4 р 4 ( ϩ 6 ) Ϫ 4 . Hence ͉ ⌬ ( x , u 4 ) ͉ Ϫ r 4 ϭ 3– Ϫ r 4 у 4 ( Ϫ 2 ) ϩ 4 Ͼ 20 . It follows that there exist 14 vertices 1 , . . . , 1 4 ⌬ ( x , u 4 ) other than y that are not adjacent to y . By Lemma 2 . 5 , the vertices i are also not adjacent to any of the vertices u 1 , u 2 and u 3 . If two vertices i and j are not adjacent , then ⌫ 1 ( x ) has a subgraph isomorphic to the first graph in Figure 2 , and if the vertices i are pairwise adjacent , then ⌫ 1 ( x ) has a subgraph isomorphic to the second graph in Figure 2 (drawing graphs a circled digit d represents a clique with d elements ; a stroke connecting this circled digit and a vertex means that every vertex of the clique is adjacent to ) . This contradicts Corollary 2 . 2 . ᮀ
L EMMA 2 . 7 . If x ϳ y , then ⌬ ( x , y ) does not contain a 3- claw .
P ROOF . Assume that ⌬ ( x , y ) contains a 3-claw ; that is , three independent vertices y , z ͖ and X : ϭ X 1 ʜ X 2 ʜ ͕ گ ( u 1 , u 2 and u 3 and a vertex z ϳ u 1 , u 2 , u 3 . Put X i : ϭ ⌬ ( x , u i X 3 . Then ͉ X i ͉ ϭ Ϫ 2 . For 1 р i Ͻ j р 3 , the vertices u i and u j have common neighbours , three of which are x , y and z ; hence ͉ X i ʝ X j ͉ р Ϫ 3 . It follows that X , at most Ϫ 4 گ ( X ͉ у 3( Ϫ 2) Ϫ 3( Ϫ 3) ϭ 3( Ϫ ϩ 1) . Since z , u 1 , u 2 , u 3 ⌬ ( x , y ͉ X , at most Ϫ 4 vertices of گ ( vertices of X are adjacent to y . Since y , u 1 , u 2 , u 3 ⌬ ( x , z X are adjacent to z . Hence if w is the number of vertices of X not adjacent to y and z , then w у ͉ X ͉ Ϫ 2( Ϫ 4) у ϩ 11 Ϫ 3 . Since ϭ 4 and ϭ 2( m Ϫ 1) or ϭ 6 and ϭ 4( m Ϫ 1) , we obtain w у 2 m Ϫ 3 Ͼ 9 .
We may therefore assume that X contains four vertices 1 , 2 , 3 and 4 that are adjacent to u 1 but not to y and not to z . Since , by Lemma 2 . 5 , the common neighbours of u 1 and u 2 and the common neighbours of u 1 and u 3 are adjacent to y , we see that u 2 and u 3 are not adjacent to the vertices 1 , 2 , 3 and 4 . If two vertices i and j are not adjacent , then ⌫ 1 ( x ) has a subgraph isomorphic to the first graph in Figure 3 , and if the vertices i are pairwise adjacent , then ⌫ 1 ( x ) has a subgraph isomorphic to the second graph in Figure 3 . This contradicts Corollary 2 . 2 . ᮀ
L EMMA 2 . 8 . If C is a maximal clique of ⌫ ( x ) , then ͉ C ͉ р ϩ 8 Ϫ 4 .
u 3 u3
z v i z u1 u 1 14 4
y v j y
u 2 u2
Ϫ–2·249141 2 . 249141 Ϫ–2·205451 2 . 205451 F IGURE 3 . Two graphs and their smallest eigenvalues . Johnson graphs and hal ed cubes 69
C , and for b B let f b be the number of گ ( P ROOF . Put c : ϭ ͉ C ͉ . Set B : ϭ ⌫ 1 ( x neighbours of b in C . Then ͉ B ͉ ϭ k Ϫ ͉ C ͉ and ͚ b B f b ϭ c ( ϩ 1 Ϫ c ) , because ͚ b B f b counts the number of pairs of adjacent vertices b B and x C and because each vertex of C has ϩ 1 Ϫ c neighbours in B . We count in two ways the number t of triples ( x , b , b Ј ) with x C , b , b Ј B and x ϳ b ϳ b Ј . On the one hand , this number is ͚ b B f b ( Ϫ f b ) . Now consider a vertex b Ј B . If x C is not adjacent to b Ј , then Lemma 2 . 4 shows that x and b Ј have at least Ϫ 2 common neighbours in ⌫ 1 ( x ) , and at least Ϫ 2 Ϫ f b Ј of these lie in B . Hence t у ͚ b Ј B ( c Ϫ f b Ј )( Ϫ 2 Ϫ f b Ј ) . Comparing the two bounds for t , we obtain ͚ b B c ( Ϫ 2) р ͚ b B f b ( Ϫ 2 f b ϩ c ϩ Ϫ 2) . Hence ͚ b B c ( Ϫ 2) р ͚ b B f b ( ϩ c ϩ Ϫ 4) . Since ͚ b B f b ϭ c ( ϩ 1 Ϫ c ) and ͉ B ͉ ϭ k Ϫ c , it follows that ( k Ϫ c )( Ϫ 2) р ( ϩ 1 Ϫ c )( ϩ c ϩ Ϫ 4) . First , consider the case in which ϭ 4 , k ϭ m 2 and ϭ 2 m Ϫ 2 . Then we obtain 2( m 2 Ϫ c ) р (2 m Ϫ 1 Ϫ c )(2 m Ϫ 2 ϩ c ) , which can be written as
2( m Ϫ 15)( m Ϫ 3) ϩ 16 р (2 m Ϫ 9 Ϫ c )(2 m Ϫ 12 ϩ c ) .
Since m у 16 , it follows that c Ͻ 2 m Ϫ 9 ϭ Ϫ 7 . Hence c р Ϫ 8 ϭ ϩ 8 Ϫ 4 . Now consider the case in which ϭ 6 , k ϭ m (2 m Ϫ 1) and ϭ 4 m Ϫ 4 . Then we obtain 4(2 m 2 Ϫ m Ϫ c ) р (4 m Ϫ 3 Ϫ c )(4 m Ϫ 2 ϩ c ) , which can be written as
4( m Ϫ 16)(2 m Ϫ 5) ϩ 92 р (4 m Ϫ 19 Ϫ c )(4 m Ϫ 22 ϩ c ) .
It follows that c Ͻ 4 m Ϫ 19 ϭ Ϫ 15 . Hence , c р Ϫ 16 ϭ ϩ 8 Ϫ 4 . ᮀ
L EMMA 2 . 9 . If x ϳ y , then ⌬ ( x , y ) does not contain three independent ertices .
P ROOF . In this proof , we use the following notation . For two subsets A and B of ⌬ ( x , y ) , we write A ϳ B if every vertex of A is adjacent to every vertex of B . For a , . ͖ x , y ͕ گ ( b ⌬ ( x , y ) , are put ⌬ Ј ( a , b ) : ϭ ⌬ ( a , b Assume that ⌬ ( x , y ) contains three independent vertices c 1 , c 2 and c 3 . For 1 р i Ͻ j р 3 , put ⌬ i j : ϭ ⌬ Ј ( c i , c j ) , and for 1 р i р 3 put
. ͖͖ i ͕ گ ͖ C i : ϭ ͕ c i ͖ ʜ ͕ z ⌬ ( x , y ) 3 z ϳ c i , z ϳ ͉ c j for j ͕ 1 , 2 , 3
Since , by Lemma 2 . 6 , ⌬ ( x , y ) does not contain four independent vertices , the sets C i are disjoint cliques . From Lemma 2 . 5 , we obtain ⌬ i j ‘ ⌬ ( x , y ) , and Lemma 2 . 7 shows that the sets ⌬ i j , 1 р i Ͻ j р 3 are pairwise disjoint . Hence
C 1 ʜ C 2 ʜ C 3 ʜ ⌬ 1 2 ʜ ⌬ 1 3 ʜ ⌬ 2 3 is a partition of ⌬ ( x , y ) . (4)
Since ͉ ⌬ i j ͉ ϭ Ϫ 2 , it follows that ͉ C 1 ͉ ϩ ͉ C 2 ͉ ϩ ͉ C 3 ͉ ϭ ϩ 6 Ϫ 3 у 24 . We proceed in several steps to derive a contradiction .
Step 1 : for i ϭ 1 , 2 , 3 , we ha e ͉ C i ͉ у 2 Ϫ 3 . We may assume that ͉ C 1 ͉ р ͉ C 2 ͉ р ͉ C 3 ͉ . Then ͉ C 3 ͉ у 8 . Put
D : ϭ ͕ d ⌬ ( x , c 1 ) 3 d ϳ ͉ y , d ϶ y ͖ .
Let a 1 , . . . , a 8 be eight vertices of C 3 and suppose that d 1 and d 2 are two vertices of D . We shall show that d 1 ϳ d 2 . Since c 1 , c 2 , and a i are independent vertices of ⌬ ( x , y ) , by Lemma 2 . 5 all of the common neighbours of two of them lie in ⌬ ( x , y ) ʜ ͕ x , y ͖ . Hence 70 K . Metsch
d 1 y 8 c1 d2
c2
F IGURE 4 . A graph with smallest eigenvalue Ϫ 2 . 154081 .
d 1 , d 2 ϳ ͉ c 2 , a 1 , . . . , a 8 . Hence , if d 1 were not adjacent to d 2 , then the graph induced by ⌫ ( x ) on ͕ y , d 1 , d 2 , c 1 , c 2 , a 1 , . . . , a 8 ͖ would be isomorphic to the graph in Figure 4 . Since , by Corollary 2 . 2 , this is not possible , it follows that d 1 ϳ d 2 . Hence D is a clique . Since every common neighbour of x and c 1 other than y lies in D or ⌬ ( x , y ) , we c 1 ͖ ) ʜ ⌬ 1 2 ʜ ⌬ 1 3 ʜ ͕ y ͖ . Since ͉ ⌬ ( x , c 1 ) ͉ ϭ and ͉ ⌬ i j ͉ ϭ Ϫ 2 , it ͕ گ have ⌬ ( x , c 1 ) ϭ D ʜ ( C 1 follows that ͉ C 1 ͉ ϩ ͉ D ͉ ϭ ϩ 4 Ϫ 2 . Since D ʜ ͕ c 1 ͖ is a clique of ⌫ ( x ) , we obtain ͉ D ͉ р ϩ 7 Ϫ 4 from Lemma 2 . 8 . Hence ͉ C 1 ͉ у 2 Ϫ 3 .
Step 2 : if c C i and j ϶ i , then c is adjacent to exactly Ϫ 2 ertices of C j ʜ ⌬ i j . W . l . o . g ., i ϭ 1 and j ϭ 2 . Since c , c 2 and c 3 are independent vertices in ⌬ ( x , y ) , we obtain ⌬ Ј ( c , c 2 ) ‘ ⌬ ( x , y ) from Lemma 2 . 5 . Thus ⌬ Ј ( c , c 2 ) ‘ ⌬ ( x , y ) ʝ ⌫ 1 ( c 2 ) ϭ c 2 ͖ ) ʜ ⌬ 1 2 ʜ ⌬ 2 3 . However , Lemma 2 . 7 implies that a vertex a of ⌬ 2 3 cannot ͕ گ C 2 ) be adjacent to c , since otherwise ⌫ ( x ) would induce a 3-claw on ͕ a , c , c 2 , c 3 ͖ . Hence ⌬ Ј ( c , c 2 ) ‘ C 2 ʜ ⌬ 1 2 . Since c ϳ ͉ c 2 and thus ͉ ⌬ Ј ( c , c 2 ) ͉ ϭ Ϫ 2 , the assertion follows .
Step 3 : if a , b ϳ C i , then a ϳ b . Since every vertex of C i is a neighbour of a and b , the vertices a and b have at least ͉ C i ͉ common neighbours . Since , by Step 1 , ͉ C i ͉ у 2 Ϫ 3 Ͼ , we must have a ϳ b .
Step 4 : if d ⌬ i j , then d ϳ C i or d ϳ C j . Assume that d ϳ ͉ C i , C j . Then there exist d i C i and d j C j with d ϳ ͉ d i , d j . Since d i ϳ ͉ c j , at most vertices of the clique C j ʜ ͕ x , y ͖ are adjacent to d i . Hence at most Ϫ 2 vertices of C j are adjacent to d i . Similarly , since d ϳ ͉ d j , at most Ϫ 2 vertices of C j are adjacent to d . Since ͉ C j ͉ Ͼ 2( Ϫ 2) by Step 1 , we can thus find d Јj C j with d Јj ϳ ͉ d , d i . Then d i , d Јj , d and c k where ͕ i , j , k ͖ ϭ ͕ 1 , 2 , 3 ͖ are four independent vertices of ⌬ ( x , y ) , contradicting Lemma 2 . 6 .
Step 5 : if ⌬ i j ϳ C i , then ⌬ i j ϳ C j . Since ͉ ⌬ i j ͉ ϭ Ϫ 2 and ⌬ i j ϳ C i , Step 2 implies that no vertex of C i is adjacent to any vertex of C j . Since by Step 2 , every vertex of C j is adjacent to Ϫ 2 vertices of C i ʜ ⌬ i j , it follows that ⌬ i j ϳ C j .
Step 6 : if i ϶ j , then at most one ertex d ⌬ i j satisfies d ϳ ͉ C i . Assume that there exist d , e ⌬ i j satisfying d , e ϳ ͉ C i . Then d and e have at most Ϫ 2 neighbours in C i . Since , by Step 2 , ͉ C i ͉ Ͼ 2( Ϫ 2) , there exists a vertex C i that is not adjacent to d and e . Step 2 shows that has at least two neighbours a 1 and a 2 in C j . Since d ϳ ͉ C i , we obtain d ϳ C j by Step 4 . Hence ͕ , c i , d , a 1 ͖ and ͕ , c i , d , a 2 ͖ are two quadrangles containing ͕ , c i , d ͖ . This contradicts Lemma 2 . 4 .
Step 7 . if i ϶ j and ⌬ i j ϳ ͉ C i , then ͉ C i ͉ ϭ ͉ C j ͉ . W . l . o . g ., i ϭ 1 and j ϭ 2 . By assumption there exists d 2 ⌬ 1 2 with d 2 ϳ ͉ C 1 . Then d 2 ϳ C 2 by Step 4 . Step 5 implies that ⌬ 1 2 ϳ ͉ C 2 , . ͖ d 1 , d 2 ͕ گ so there exists d 1 ⌬ 1 2 with d 1 ϳ ͉ C 2 . Then d 1 ϳ C 1 by Step 4 . Put ⌬ : ϭ ⌬ 1 2 Then ͉ ⌬ ͉ ϭ Ϫ 4 and Step 6 shows that ⌬ ϳ C 1 , C 2 . Thus Step 3 implies that d 1 , d 2 ϳ ⌬ . Consider a 1 C 1 , with a 1 ϳ ͉ d 2 . Since ͉ ⌬ 1 2 ͉ ϭ Ϫ 2 , Step 2 shows that a 1 ϳ a 2 for some vertex a 2 C 2 . Hence ⌬ Ј ( a 1 , c 2 ) ϭ ͕ d 1 , a 2 ͖ ʜ ⌬ . Hence ⌬ Ј ( a 1 , d 2 ) ϭ ͕ c 1 , a 2 ͖ ʜ ⌬ . Since d 1 ϳ a 1 , it follows that d 1 ϳ ͉ d 2 . Hence ⌬ Ј ( d 1 , d 2 ) ϭ ͕ c 1 , c 2 ͖ ʜ ⌬ . Consider b 1 c 1 ͖ . Then b 1 ϳ d 1 , so b 1 ϳ ͉ d 2 . Hence b 1 is adjacent to all vertices of ⌬ 1 2 except ͕ گ C 1 Johnson graphs and hal ed cubes 71
͖ c 2 ͕ گ d 2 . Thus , by Step 2 , b 1 has a unique neighbour in C 2 . Similarly , every vertex of C 2 has a unique neighbour in C 1 . Hence ͉ C 1 ͉ ϭ ͉ C 2 ͉ .
Step 8 : if d 2 ⌬ 1 2 and d ⌬ ( x , c 1 ) such that d 2 ϳ ͉ C 1 and d ϳ ͉ y , then d 2 ϳ d . Since d ϳ ͉ y , at most vertices of the clique C 1 ʜ ͕ x , y ͖ can be adjacent to d . Similarly , since d 2 ϳ ͉ C 1 , at most vertices of C 1 ʜ ͕ x , y ͖ can be adjacent to d 2 . Since d ϳ x , c 1 and d 2 ϳ x , y , c 1 , it follows that at most 2( Ϫ 2) vertices of C 1 are adjacent to d or to d 2 . Since , by Step 1 , ͉ C 1 ͉ Ͼ 2( Ϫ 2) , we can thus find a vertex c C 1 with c ϳ 3 d , d 2 . Assume that d 2 ϳ ͉ d . Then d , c and d 2 are three independent vertices of ⌬ ( x , c 1 ) , so ⌬ ( c , d 2 ) ‘ ⌬ ( x , c 1 ) ʜ ͕ x , c 1 ͖ by Lemma 2 . 5 . Since c ϳ ͉ d 2 , Step 2 shows that c has a neighbour c Ј in C 2 . Since d 2 ϳ C 2 , we have c Ј ϳ d 2 . Hence c Ј ⌬ ( c , d 2 ) but c Ј ԫ ⌬ ( x , c 1 ) ʜ ͕ x , c 1 ͖ , since c Ј ϳ ͉ c 1 . This is a contradiction .
Step 9 : either C 1 ϳ ⌬ 1 2 or C 1 ϳ ⌬ 1 3 . Assume that there exist d 2 ⌬ 1 2 and d 3 ⌬ 1 3 with d 2 , d 3 ϳ ͉ C 1 . Then ͉ C 1 ͉ ϭ ͉ C 2 ͉ ϭ ͉ C 3 ͉ by Step 7 . Using (4) , we obtain ͉ C 1 ͉ ϭ ( ϩ 6 Ϫ 1– 3 ) / 3 ϭ 3 ϩ 2 Ϫ . Put
D : ϭ ͕ d ⌬ ( x , c 1 ) 3 d ϳ ͉ y , d ϶ y ͖ .
As in Step 1 , we see that D is a clique and that ͉ C 1 ͉ ϩ ͉ D ͉ ϭ ϩ 4 Ϫ 2 . Hence 2– ͉ D ͉ ϭ 3 ϩ 2 Ϫ . By Step 8 , every vertex of D is adjacent to d 2 . The same argument shows that every vertex of D is adjacent to d 3 . Similarly , if
D Ј : ϭ ͕ d ⌬ ( y , c 1 ) 3 d ϳ ͉ x , d ϶ x ͖ ,
2– then ͉ D Ј ͉ ϭ 3 ϩ 2 Ϫ and every vertex of D Ј is adjacent to d 2 and d 3 . Hence 4– D ʜ D Ј ʜ ͕ x , y , c 1 ͖ ‘ ⌬ ( d 2 , d 3 ) . Since D ʝ D Ј ϭ C , we conclude that ͉ ⌬ ( d 2 , d 3 ) ͉ у 3 ϩ 1– 7 Ϫ 2 . Since ͉ ⌬ ( d 2 , d 3 ) ͉ р , it follows that 3 р 2 Ϫ 7 , a contradiction . Now we are in position to derive a contradiction . In view of Step 9 , we may assume
that C 1 ϳ ⌬ 1 2 . Step 5 implies that C 2 ϳ ⌬ 1 2 . Similarly , we have C 3 ϳ ⌬ 1 3 or C 3 ϳ ⌬ 2 3 , so that we may assume that C 3 ϳ ⌬ 1 3 . By Step 5 , C 1 ϳ ⌬ 1 3 . Hence C 1 ϳ ⌬ 1 2 , ⌬ 1 3 , C 2 ϳ ⌬ 1 2 and C 3 ϳ ⌬ 1 3 . Step 3 shows that ⌬ 1 2 ϳ ⌬ 1 3 . Consider d ⌬ 2 3 . In view of Step 4 , we may assume that d ϳ C 2 . Consider e ⌬ 1 2 . Then d , e ϳ C 2 , so d ϳ e by Step 3 . Hence d ⌬ ( e , c 3 ) . Since ⌬ 1 2 ϳ ⌬ 1 3 , we have ⌬ 1 3 ‘ ⌬ ( e , c 3 ) . Since also x , y ⌬ ( e , c 3 ) , it follows that e and c 3 have at least ͉ ⌬ 1 3 ͉ ϩ 3 ϭ ϩ 1 common neighbours . Hence e ϳ c 3 . Thus e ϳ c 1 , c 2 , c 3 , which means that e lies in all sets ⌬ i j , 1 р i Ͻ j р 3 . This contradicts (4) . ᮀ
L EMMA 2 . 10 . Consider a ertex x and a maximal independent subset A of ⌫ 1 ( x ) . Then ͉ A ͉ ϭ m , any two ertices of A ha e Ϫ 2 common neighbours in ⌫ ( x ) , and e ery ertex of ⌫ ( x ) that does not lie in A has two neighbours in A .
A that have i گ ( P ROOF . Put a : ϭ ͉ A ͉ and denote by f i the number of vertices of ⌫ 1 ( x neighbours in A . Then f 0 ϭ 0 , since A is maximal , and f i ϭ 0 for i у 3 , since ⌫ ( x ) does not contain a 3-claw (Lemma 2 . 9) . Hence f 1 ϩ f 2 ϭ ͉ ⌫ 1 ( x ) ͉ Ϫ a ϭ k Ϫ a . Since each vertex of A has neighbours in ⌫ ( x ) , we have f 1 ϩ 2 f 2 ϭ a . Hence f 2 ϭ a ( ϩ 1) Ϫ k . By Lemma 2 . 4 , any two vertices of A have at least Ϫ 2 common neighbours in ⌫ ( x ) . It 1– –1 follows that f 2 у 2 a ( a Ϫ 1)( Ϫ 2) . Hence 2 a ( a Ϫ 1)( Ϫ 2) р a ( ϩ 1) Ϫ k , with equality if f any two vertices of A have exactly Ϫ 2 common neighbours in ⌫ ( x ) . First consider the case ϭ 4 . Then k ϭ m 2 and ϭ 2( m Ϫ 1) . We obtain a ( a Ϫ 1) р a (2 m Ϫ 1) Ϫ m 2 . Hence ( a Ϫ m ) 2 р 0 . It follows that a ϭ m and we have equality . Now consider the case ϭ 6 . Then k ϭ m (2 m Ϫ 1) and ϭ 4( m Ϫ 1) . We obtain 2 a ( a Ϫ 1) р a (4 m Ϫ 3) Ϫ m (2 m Ϫ 1) . 72 K . Metsch
1– This implies that 2( a Ϫ m )( a Ϫ m ϩ 2 ) р 0 . Hence a ϭ m and we have equality . Hence , in both cases , we have a ϭ m and equality . It follows that any two vertices of
A have exactly Ϫ 2 common neighbours in ⌫ ( x ) . Also f 2 ϭ a ( ϩ 1) Ϫ k ϭ m ( ϩ 1) Ϫ k ϭ k Ϫ a , so that f 1 ϭ 0 . Hence every vertex of ⌫ ( x ) that is not in A is adjacent to exactly two vertices of A . ᮀ
L EMMA 2 . 11 . Consider a ertex x , two non - adjacent neighbours y and z of x , and let ⌬ be the set of common neighbours of y and z in ⌫ ( x ) . Then ͉ ⌬ ͉ ϭ Ϫ 2 . If ϭ 4 , then the two ertices of ⌬ are not adjacent . If ϭ 6 , then the ertices of ⌬ form a 4- cycle .
P ROOF . Let A be any maximal independent subset of ⌫ 1 ( x ) containing y and z . Then Lemma 2 . 10 shows that ͉ A ͉ ϭ m and ͉ ⌬ ͉ ϭ Ϫ 2 . Consider a vertex u ⌬ and put y , z ͖ . Then B Ј is an independent set . Let B be a maximal independent ͕ گ ( ͖ B Ј : ϭ ( A ʜ ͕ u subset of ⌫ 1 ( x ) containing B Ј . Lemma 2 . 10 shows that ͉ B ͉ ϭ m and that every vertex of B has two neighbours in B . Let be the vertex with B ϭ B Ј ʜ ͕ ͖ . Then ԫ A , so گ ( x ) ⌫ . B گ has two neighbours in A . Since B , these must be the two vertices y and z of A Hence ⌬ and u ϳ ͉ . The same argument shows that every vertex of ⌬ is not adjacent to some other vertex of ⌬ . If ϭ 4 , then ⌬ ϭ ͕ u , ͖ and the proof is complete . Now consider the case in which ϭ 6 . Let w 1 and w 2 be the two vertices of ⌬ other than u and . Then y and z are the two neighbours of w i in A , so the two neighbours of w i in B must be the two elements u , ⌬ A . Hence w 1 , w 2 ϳ u , . Since w 1 is not adjacent to some other vertex of گ and of B we must have w 1 ϳ ͉ w 2 . Hence ⌬ is a 4-cycle . ᮀ
L EMMA 2 . 12 . If ϭ 4 , then ⌫ ( x ) is an m ϫ m - grid for e ery ertex x .
P ROOF . In this proof , we call a maximal clique of ⌫ ( x ) a line . Consider two adjacent vertices y and z of ⌫ ( x ) . It follows from Lemma 2 . 11 that any two vertices of ⌫ ( x ) that are adjacent to y and z are adjacent . Hence y and z lie in a unique line , which consists of y and z and all common neighbours of y and z in ⌫ ( x ) . Consider y ⌫ ( x ) . Since ⌬ ( x , y ) is not a clique (Lemma 2 . 8) , we see that y lies in at least two lines . Lemma 2 . 9 implies that y lies in at most two lines . Hence y lies in exactly two lines l 1 and l 2 . We have ͉ l 1 ͉ ϩ ͉ l 2 ͉ ϭ ͉ ⌬ ( x , y ) ͉ ϩ 2 ϭ 2 m . Now consider a vertex z of ⌫ ( x ) not on l 1 or l 2 . Then z ϳ ͉ y , so y and z have two common neighbours in ⌫ ( x ) and these are non-adjacent (Lemma 2 . 11) . It follows that z has a unique neighbour in each of the lines l 1 and l 2 . Since z also lies on two lines , it follows that one line on z meets l 1 and misses l 2 and the other line on z meets l 2 and misses l 1 . Since this holds for any two non-adjacent vertices y and z of ⌫ ( x ) , it follows that there are integers r , s у 2 with r ϩ s ϭ 2 m such that the lines form an ( r ϫ s )-grid on the vertices of ⌫ ( x ) . Since ⌫ ( x ) has m 2 vertices , we must have rs ϭ m 2 . Hence r ϭ s ϭ m . ᮀ
L EMMA 2 . 13 . If ϭ 4 , then ⌫ is the folded Johnson graph J (2 m , m ) .
P ROOF . Consider two vertices u and at distance two , and a vertex x ϳ u , . By Lemma 2 . 11 , there exist two non-adjacent common neighbours y and z of u and i n ⌫ ( x ) . Let x Ј be the remaining common neighbour of u and . Then x Ј ϳ ͉ x . Hence y and z must be the two vertices of ⌫ ( x Ј ) adjacent to u and . Consequently , the four common neighbours x , y , x Ј and z of u and form a 4-cycle . It follows from a result of Johnson graphs and hal ed cubes 73
2 m Blokhuis and Brouwer [2] that ⌫ is a quotient of the Johnson graph ( m ) . It follows from the parameters that ⌫ must be the folded Johnson graph . ᮀ
L EMMA 2 . 14 . Suppose that ϭ 6 and consider adjacent ertices x and y . Then ⌬ ( x , y ) contains exactly two maximal cliques C with ͉ C ͉ у 14 . Furthermore , ⌬ ( x , y ) is the disjoint union of these two cliques .
P ROOF . In view of Lemma 2 . 8 , we can find non-adjacent vertices c 1 , c 2 ⌬ ( x , y ) . Put
⌬ : ϭ ͕ c ⌬ ( x , y ) 3 c ϳ c 1 , c 2 ͖ ,
C 1 : ϭ ͕ c ⌬ ( x , y ) 3 c 1 ϳ c ϳ ͉ c 2 ͖ ʜ ͕ c 1 ͖ , and
C 2 : ϭ ͕ c ⌬ ( x , y ) 3 c 1 ϳ ͉ c ϳ c 2 ͖ ʜ ͕ c 2 ͖ . Since ⌫ ( x ) does not contain a 3-claw , ⌬ ( x , y ) does not contain three independent vertices . It follows that the sets C 1 and C 2 are cliques and that C 1 ʜ C 2 ʜ ⌬ is a partition of ⌬ ( x , y ) . Since , by Lemma 2 . 11 , the common neighbours of c 1 , c 2 and x form a 4-cycle , there exist non-adjacent vertices d 1 and d 2 such that ⌬ ϭ ͕ d 1 , d 2 ͖ . Hence ͉ C 1 ͉ ϩ ͉ C 2 ͉ ϭ Ϫ 2 . Lemma 2 . 8 shows that ͉ C i ͉ р Ϫ 16 . Hence ͉ C i ͉ у 14 . Since ϭ 6 , two distinct maximal cliques of ⌬ ( x , y ) can have at most Ϫ 2 ϭ 4 vertices in common . This implies that C i lies in a unique maximal clique C i of ⌫ ( x , y ) . Consider any maximal clique C of ⌬ ( x , y ) with at least 14 vertices . Since C 1 ʜ C 2 ʜ ⌬ is a partition of ⌬ ( x , y ) , it follows that C shares at least six elements with C 1 or C 2 . Thus C ϭ C 1 or C ϭ C 2 . Since d 1 ϳ ͉ d 2 , we can repeat the above argument to construct maximal cliques D i with ͉ D i ͉ у 14 and d i D i . Since C 1 and C 2 are the only maximal cliques of ⌫ ( x ) with at least 14 vertices , we must have ͕ D 1 , D 2 ͖ ϭ ͕ C 1 , C 2 ͖ . Hence d 1 , d 2 C 1 ʜ C 2 . It follows that C 1 ʜ C 2 ϭ ⌬ ( x , y ) . Since d 1 ϳ ͉ d 2 and C 1 ʝ C 2 ‘ ⌬ ( c 1 , c 2 ) ʝ ⌬ ( x , y ) ϭ ⌬ ϭ ͕ d 1 , d 2 ͖ , we have C 1 ʝ C 2 ϭ C . ᮀ
L EMMA 2 . 15 . If ϭ 6 , then ⌫ ( x ) is the triangular graph T (2 m ) .
P ROOF . Call the maximal cliques of ⌫ ( x ) with at least 15 vertices lines . By the preceding lemma , every vertex of ⌫ ( x ) lies in two lines , and two adjacent vertices of ⌫ ( x ) occur in a unique line .
Consider distinct lines l 1 and l 2 . Then we can find vertices 1 l 1 and 2 l 2 with 1 ϳ ͉ 2 . Since , by Lemma 2 . 11 , the vertices 1 and 2 have four common neighbours in ⌫ ( x ) and since lines can meet in at most one point , it follows that each of the two lines on 1 meets each of the two lines on 2 . Hence l 1 meets l 2 in a unique point . The same argument shows that any two distinct lines meet in a unique point . Hence , if l is a line , then , since every vertex of l lies in a unique second line , the number of lines is ͉ l ͉ ϩ 1 . It follows that all lines have the same size s . Since the two lines on a vertex consist of and the common neighbours of x and , we have 2( s Ϫ 1) ϭ . Hence s ϭ 2 m Ϫ 1 and there exist 2 m lines . Since distinct lines meet and since points lie on two lines , the structure consisting of the vertices and lines is dual to the complete graph on 2 m vertices ; that is , ⌫ ( x ) is the triangular graph T (2 m ) . ᮀ
L EMMA 2 . 16 . If ϭ 6 , then ⌫ is the folded hal ed cube of diameter d . 74 K . Metsch
P ROOF . Using the preceding lemma , the conclusion follows as in Step 11 of Theorem 3 . 3 in [4] . ᮀ
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Recei ed 9 October 1 9 9 5 and accepted in re ised form 1 3 February 1 9 9 6
K LAUS M ETSCH
* Mathematisches Institut , Arndtstrasse 2 , D - 3 5 3 9 2 Giessen , Germany