A Bernstein theorem for special Lagrangian graphs by Jost J., Xin Y. L.

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By Jost J., Xin Y. L.

We receive a Bernstein theorem for distinctive Lagrangian graphs in for arbitrary basically assuming bounded slope yet no quantitative restrict.

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As a consequence of t h i s p r e l i m i n a r y v e r s i o n of t h e main r e s u l t i n t h e proof o f t h e main theorem w e may assume t h a t M = 0 ( n l o g n ) Theorem 3. If In log n < M i N In log n then uM LM/n! 0 . - - . B. Bollobds 38 Now we are r e a d y t o p r o v e t h e fundamental theorem o f Wright C101. Theorem 4 . Suppose + #n(log n Then UM $(n)) M 5 and + m - N 5 + #n(log n . Jl(n)) . LM/n! N $(n) The map s e n d i n g a g r a p h i n t o i t s complement p r e s e r v e s t h e Proof.

When n = 3, the only unicyclic graph is the triangle K3 , so t r = 3. There a r e j u s t two u n i c y c l i c g r a p h s w i t h n = 4 , b o t h having t r = 4 . A connected g r a p h w i t h Among t h e f i v e u n i c y c l i c g r a p h s G with n = 5 , a l l b u t one have t r ( G ) = 5 , t h e e x c e p t i o n b e i n g t h e graph o f F i g u r e l a which h a s t r = 4 , so t h a t p(0,5) = 4. There a r e e x a c t l y 10 u n i c y c l i c g r a p h s of o r d e r 6 , j u s t two o f which ( F i g u r e l b , c ) a t t a i n t h e s m a l l e s t The trail number of a graph 53 possible trail number, 4.

F (d) Figure 2 . 2 26 J. Bermond, C. Delorrne and G. 3 P G w i l l be s a i d t o s a t i s f y p r o p e r t y P , i f any p a i r o f v e r t i c e s a t d i s t a n c e D (where D i s t h e d i a m e t e r o f G) c a n be j o i n e d by a p a t h o f l e n g t h D + l A t r i v i a l example i s g i v e n by t h e complete g r a p h Kn,n 2 3 : o t h e r examples w i l l be g i v e n i n s e c t i o n 4 . A graph . THE M A I N THEOREM I11 G be a graph o f diameter D 2 1 , s a t i s f y i n g property G' be a graph s a t i s f y i n g p r o p e r t y P* ( w i t h i n v o l u t i o n Then G*G' , w i t h f ( x , y ) ( x ' ) = f ( x ' ) f o r e v e r y arc ( x , y ) f) of an a r b i t r a r y o r i e n t a t i o n of G , i s a graph of diameter a t m o s t Theorem: P Let and l e t .

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