Cycles in Graphs by B. R. Alspach

By B. R. Alspach

This quantity bargains with numerous difficulties concerning cycles in graphs and circuits in digraphs. major researchers during this quarter current the following three survey papers and forty two papers containing new effects. there's additionally a suite of unsolved difficulties.

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3) H(d) . 11), we have r = m ord(x) - n e and Substitution of these expressions for yields: s r = and IG : i x > ( n . Curran and D. Witte 54 m o r d ( x ) - n e + IG : < x > I n < \GI ; (m o r d ( x ) - n e ) t 5 ( ( G : <*In) ( p - t ) ; (m o r d ( x ) - n e ) ( t + l ) (IG :

A u s t r a l . Math. Seymur, Sums o f c i r c u i t s , Graph theory and r e l a t e d t o p i c s , ed. A. R. Murty, (Academic P r e s s , New York, 1 9 7 9 ) , 341-355. , 8 (19731, 367-387. V. D. A. I n t h i s p a p e r i t i s shown t h a t e v e r y c o n n e c t e d metac i r c u l a n t g r a p h h a v i n g a n e v e n number o f b l o c k s o f prime c a r d i n a l i t y , o t h e r t h a n t h e sole e x c e p t i o n o f t h e P e t e r s e n graph, h a s a Hamilton cycle. T h i s p a p e r i s a s e q u e l t o [31 in which i t was shown t h a t e v e r y c o n n e c t e d m e t a c i r c u l a n t g r a p h w i t h a n odd number o f v e r t i c e s g r e a t e r t h a n o n e a n d w i t h For purposes of b r e v i t y , we b l o c k s of p r i m e c a r d i n a l i t y h a s a H a m i l t o n c y c l e .

Witte Figure 2. k A simple case: Ak G = Z 5 X 2 X 2 4' Uk hk 0 3 5 7 11 13 15 19 Figure 3. A simple case: d d 0 10 1 1I. 2 12 3 13 4 14 5 15 6 16 7 17 8 18 9 19 PROPOSITION 3 . 1 5 . The nwnber of s o l u t i o n s (with 0 5 t c IG nwnber of l a t t i c e points Proof. which I : I t = O - 1 (ft -1) . 4 ' of (t,d) Bt(d) = ( 0 , O ) i s one l e s s than t h e 0 5 d < ord(y-x)) i n t h e closed triangZe i: 5 T which s a t i s f y a r e t h e v a l u e s of (t,d) of gcd(A) = 1 , d for Bt(d) = (0,O) Hamilton Paths in Curtesian Products For each t , are precisely A the l a t t i c e points A~( t ) , .

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