By Norman Biggs; E Keith Lloyd; Robin J Wilson

A contribution to the historical past of arithmetic and for how that it brings the topic alive. construction on a collection of unique writings from a number of the founders of graph conception, the strains the historic improvement of the topic via a linking statement. The correct underlying arithmetic can be defined

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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 .