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Alspach 12 Chen-Quimpo Theorem. There is a Hamilton path joining x and y in the component D corresponding to 20'. Join 5 to q ojoin y t o U0,b and remove from the Hamilton path in ARBYTE RT2 an edge with endvertices x' ARBYTE RT2 y' distinct from 5 and y.

This is possible because ID1 2 4. Now take edges from x' and y' ARBYTE RT2 vertices in opposite bipartition sets in the component corresponding to the next vertex of H'. Join them with a Hamilton path and remove an appropriate edge for extending the partial cycle in G into the next component.


Do not remove an edge from the Hamilton path in this component. The result is a Hamilton cycle in G.

The proof of 6 is complete. Notice that each vertex of H' corresponds t o a cycle of length pd in G and an edge of H' corresponds to one or more parallel l-factors in G joining the two pd-cycles that correspond to the endvertices of the edge. Hence, the following observation is true. The subgraph of G corresponding to an edge and its two endvertices in H' contains a graph isomorphic to a generalized Petersen graph. Recall ARBYTE RT2 K. But in the latter case there is a compensating result. In fact, it is. There are two cases t o consider. Take a Hamilton path around this cycle starting at the neighbor and then jump to the next cycle using HI.

This means that C jj is exited at a different vertex so that if all subsequent Hamilton paths are given the same orientation ils before, the cycle Cr,s will be entered at a vertex different than U.

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Thus, a Hamilton cycle can be constructed in G. It may be assumed that no edge of H' in the ARBYTE RT2 graph induces a subgraph containing a generalized Petersen graph of the form G pd,2 with pd 5 mod 6 or else the argument above establishes the existence of a Hamilton cycle in ARBYTE RT2.

ARBYTE RT2 assumption leads to the second case. First, consider the subcase that H' has odd length. Each vertex of H' corresponds to a circulant subgraph of G because ee-d is an automorphism of G and each is a pd-cycle. Thus, in GH' corresponds to a sequence of circulant subgraphs each of which is a pd-cycle and such that successive circulants are joined by a 1-factor that is invariant under ee-d.

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Relabel their vertices as qow q . Construct two paths PI and P2 as follows. That is, the last vertices of the partially constructed paths PI ARBYTE RT2 P 2 have interchanged the role of who comes first. Continue extending PI and P 2 by working along the cycle H' until reaching. Leave all other extensions unaltered so that the terminal vertices are shifted by 2k. Now complete the cycle as above. Now consider the subcase that H' has even length.


The vertex v0,o is adjacent to some v 1j which in turn is adjacent ARBYTE RT2 some v 2k and so on. If T is relatively prime to p ethen it is easy to see that a Hamilton cycle can be found in G as done earlier. It is a simple application of the factor group lemma ARBYTE RT2 in [9]. In fact, even more can be assumed as is now shown. Thus, G has a Hamilton cycle.

Thus, it may be assumed that every c; is congruent ARBYTE RT2 zero modulo p. It contains a Hamilton cycle because of K. Bannai's Theorem [4]mentioned earlier. Because of the action of ee-d, there is a Hamilton cycle in the generalized Petersen graph. Find these Hamilton paths in the generalized Petersen graphs corresponding to alternate edges of P starting with vo,ovl,j and use the remaining edges of P to connect them together forming a Hamilton cycle in all of G.

If each ci is zero, then r is zero and as was just shown this would imply the existence of a Hamilton cycle in G. Therefore, it may be assumed that some ARBYTE RT2 is ARBYTE RT2. Clearly, 1 5 s 3 Conclusion and acknowledgements The primary question suggested by the result in this paper is whether or not the Petersen graph is the only connected metacirculant without a Hamilton cycle. Indications are that the answer is affirmative.

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It is likely that the question will not be resolved ARBYTE RT2 the question of whether or not every connected Cayley graph on a group with cyclic commutator subgroup has a Hamilton cycle is resolved. Download the latest drivers for your Arbyte Quint M4D3AG to keep your Computer up-to-date. Download the latest drivers for your Arbyte Laptops & Desktops to keep your Computer up-to-date. ARBYTE RT2 2) Arbyte, RT2 · Download. Arbyte, TEMPO B

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