# Cycles in Graphs by B. R. Alspach

By B. R. Alspach

This quantity offers with various difficulties concerning cycles in graphs and circuits in digraphs. prime researchers during this zone current the following three survey papers and forty two papers containing new effects. there's additionally a set of unsolved difficulties.

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The c a s e and i s shown i n F i g u r e 3 . G(11,Z) starting with l l ~ 3 is done. Successive substitu- i = 7 i n F i g u r e 3 g i v e H a m i l t o n p a t h s from f o r a l l n 2 1 7 , n :5 (mod 6 ) . Now l e t n = mt2 n- 2 s u c c e s s i v e t y p e D i n s e r t i o n s g i v e Hamilton p a t h s from uo t o uo n to ? v mt8, n 5 (mod 6 ) . A Hamilton v for all m A H a m i l t o n p a t h from u m : 5 (mod 6 ) . Finally, let u10 and t h e n 0 to uO u2 u3 Figure 3 u5 G(11,Z) in i s shown i n F i g u r e 4 .

X So = 0 # 0 and some . i Since # 0 G . i s c o n n e c t e d , w e must h a v e t h a t = x t o g e t h e r imply t h a t lSil am = 1. T h i s i m p l i e s t h a t Cayley g r a p h by i n v o k i n g Theorem 9 o f 121. - a m- 1 Case 2. x C Si 0 so t h a t the r e s u l t i n g modified path i i n t h e same way as i n t h e p r e c e d i n g p a r a g r a p h . that = i n the a 2 u m-l and t e r m i n a t e s a t 0 P i a 2-a 3-. -a ... -a uo to i n-1 . G i -a '-a Finally, i n the case t h a t i 3 L i n k i n g t h i s Hamilton p a t h t o o b v i o u s manner y i e l d s a H a m i l t o n c y c l e i n i 33 The g r o u p o v e r which G = 1.

Denote The rnetacircuZant graph {aa: a € A}. S. Zn m u l t i p l i c a t i v e group of u n i t s i n n ) 11 ui and n a € A) , and has vertex is adjacent t o t h a t the following hold: , and . *This r e s e a r c h w a s p a r t i a l l y s u p p o r t e d b y t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada u n d e r G r a n t A-4792. **This r e s e a r c h was p a r t i a l l y s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n under G r a n t MCS-80002263.