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By Gunter Schaar

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K}, p(l) * P (2) • If dg (i1 #i2 ) = 1 and j 1 «j2 are isolated vertices in G 2 then a (Q 1 ,Q2 )-path of length two needs not exist in G* We see that in case (i) for each pa i r m,n € £ i f.. } and £rJ2,r J} are edges of G of the form £(i, j), (i**j *)} with i = i* and [j,j*} £ E ( G 2 ). k}. J) . (i 1 ,3")} with i = i* and f j J e E(G2 ). (i,J*))-paths of length 1 ’ , 3 * 1 * * |V(G)I - |V(Hp )| + 1, no inner ve rt ex of which is belonging to Hp . ) Thus we obtain ( Q ^ S ^ - p a t h s of length 1 with |V(Hp )l + 1 < 1 * I V ( G )1 - 1.

I,t)), i ■ wher e the first components of the vertices are taken mod s. We obtain immediately that these sequences are edge-sequences and that c± :* (w 1 #w 2 ,.. ,w^# (s,i)) are two Hamiltonian cycles in G; it remains to show E ( c ^ ) a As su mi ng the opposite e (c 2 ) = 0. we have an edge e* |(i 1 #J ^ ) * (i2 #J 2 )}€E(c 1 )nE(c2 ). t}). We consider case (a). Because of e E E f c ^ ) we get i^ s ±2+l mod s, and from e 6 E(c2 ) it follows i^ = i 2-l BO<* s# Hence 2 3 0 mod s, but this is not possible since s ^ 3.

Let H be a graph with s vertices, G ^ , • • •,GS pairwise vertex-disjoint graphs and V a numbering of the vertices of H, i«e« V : i 1 ,. •. ,s } ^ k I— e V(H) is a bi- jective mapping of {l,«««,s} onto V(H). The loin of G^ «. « »Ga over H,y is defined to be the graph 3 = 3 ( H , v ;G^,««*,Gs ) with V(3) :» U k=i E(0) s- V(G. ), K {f x. y} s x€V(G1 ) A y ( ( i = j A { x . y } 6 E ( Gi )) v 6 V ( Gj ) { V t , Vj} For the special case H = K g the graph 3 ( K8 , v € A E(H))| . ,G3 ). 2, proof of Theorem 21.