Hamiltonian properties of products of graphs and digraphs by Gunter Schaar
By Gunter Schaar
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Additional info for Hamiltonian properties of products of graphs and digraphs
This result, Theorem 1 and Theorem 18(a) are the only known conditions which are both necessary and sufficient for some of the five classical products to be traceable or Hamiltonian. For a connected graph G a closed edge-sequence c is called a cover o f G iff each v e rt ex of G is contained in c. A q- cover of G is a cover c of G passing each ve rt ex of G exactly q times, q = 1,2,... ) An (i^,. ^ is passed exactly once. Every cover c of G can be uniquely decomposed with respect to a fixed vertex i into (i)-covers c p of some induced sub graphs G S!
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. N o w we consider the case 1 = | V ( H p )|. Obviously, because of |V(Hp )| i | V(G)| - 1 there is a subgraph H q 6 of G which is different from Hp . d* )} with f J . J ’/ C E(G2 ). ,Sf . For a c e rt ai ny U€ |l ,. ,s -1} we have K i . J M i .
Then G ^ G g S H G t-cover c # of iff (*) there is a and there is an s-cover c" of G 2 , such that for all vertices i € V ( G ^ ) , j € V ( G 2 ) and integers m i m 1 ^2, n ^ n ^ 2 the following implication holds: If m X i(4 ) k- « 1 z r=ni i(4) . 17) then Z. k=l l(c’) * K Z r=l l(c" ) r . 18) (Id »Jd ) • in G ^ x G 2 where d - st. 17) for any vertices i € V(G^) and J € V ( G 2 ). 19) . From (2*16) and (2*19) it follows that the 1-cover c of G ^ x g 2 must pass the ve rt ex (i»j) at least twice» which is a contradiction.