By B. R. Alspach
This quantity offers with various difficulties related to cycles in graphs and circuits in digraphs. prime 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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Additional info for Cycles in Graphs
2. rx + sy Embed Cay(x,y : G ) Cay(x,y : G ) of in the torus. 3. knot(C) E Z Cay(x,y : G ) torus: Lift C in Cay(x,y : G) , we let C considered as an oriented knot on the to some path in the plane. of this path, and let in the natural way. For any elementary circuit C Z be the knot class of x on the torus by identifying a vertex with the point Let (a,b) be the initial endpoint (c,d) be the terminal endpoint. Then knot(C) = ( c - a , d-b). Intuitively, if times knot(C) = (m,n) , then m is the (algebraic) number of C wraps around the torus longitudinally, and n is the number of times C wraps around the torus meridionally [9, pp.
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 .
THEOREM 7 . 1 . 9(b), and o n l y i f , f o r some t , t h e digraph 0 to h a s knot class Ht(d) Theorem 7 . 1 i s a s p e c i a l case of t h e f o l l o w i n g r e s u l t . Hamilton p a t h i f and o n l y i f i t s knot c l a s s i s (0,O) - d(y-x) (0,O) Because . x if So is a Ht(d) (Lemma 5 . 8), t h i s r e s u l t a l s o g i v e s a Cay(x,y : G ) c h a r a c t e r i z a t i o n of t h e Hamilton p a t h s i n The knot class o f THEOREM 7 . 2 . and with d 0 5 t c IG :c y - x i l . H (d) i s equal t o Bt(d) t 0 5 d c ord(y-x) .
Cycles in Graphs by B. R. Alspach