By B. R. Alspach

This quantity bargains with numerous difficulties related to 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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I f any p a i r Lemma 9 w i t h {Ci,Ci+l}, i = 1,3,5, are v e r t e x - G . Thus, e a c h p a i r by j o i n i n g e a c h { C G where i n d i c e s are reduced modulo 6 . We assume t h a t Co , . ,C5 are n o t a l l d i s t i n c t . = C2 without l o s s of g e n e r a l i t y . I f e i t h e r C4 o r 0 then w e could g e t a d o u b l e c y c l e c o v e r of G by drawing t h e {elye3,e5} Co This t h e n t h e same scheme j u s t employed f o r two v e r t e x - P = (ei), double c y c l e cover of from .

Let C1 and algebraic i n t e r s e c t i o n number of C1 crosses C2 45 C2 be oriented knots on the torus. The Cl with from left to right. C2 is the algebraic number of times See [8, Section 68G, pp. 231-2321 for a rigorous definition of this notion. [9, p. 281. 9. 10. If Cl and , then Cay(x,y : G) C2 are d i s j o i n t elementary c i r c u i t s i n knot(C1) = knot(C2) . 9 implies m n = nlm2. Since Proof. Let and gcd(ml,nl) nl = = fn2. 11. Let H be any subdigraph of no two elementary circuits in H Cay(x,y two elementary c i r c u i t s i n (a) Then gcd(knot(H)) circuits i n (b) Let H = H Zc knot(C) , in H .

P. We can The s e t of draws t h e c h o r d a l p a t h s as desired. DEFINITION. chain i n in X X Let be a c o l l e c t i o n of c y c l e s and linking a path X w i t h a set . P f z if U P C U D of c y c l e s i n Z iC,Dl join ' I LEMMA 8. Let = Proof. Both is cubic, G and C U D U P linking a X U D u P) i s isomorphic t o of R(C U D) respectively. C U D U P and p' J(R(C (p')-'(f) (u,v)-path via {C,D} P . are r e d u c i b l e . P and Let p W C U and u D only a t u and v where f and g D),f,g) (p')-'(g) A s i n t h e proof o f Lemma 6 , contain p'(C A D) u G = C .