Several Sufficient Conditions Of Hamilton-Connected Graphs

This paper gives four sufficient conditions for l-Hamilton-Connected graphs and Almost-Hamilton-Connected graphs by the vertex inserting lemmas and LTW-sequence in [2].These conditons are connected with the neighborhood insertsections and unions of the independent essential sets which belong to Ik+1(3)(G), Ik+1(G*) and Ik(3) (G) respectively.Theorem 1 Let G be a (k + s + 2)-connected graph of order n with k > 2. and be a k-LTW-sequence. If aiSi(X) > n + s for each X Ik+1(G*).then G is s-Hamilton-Connected.Theorem 2 Let G be a (k + 2)-connected graph of order n with k > 2. b be an interger and 0 < b < k.b*= min{k, (2b-1+k)/2}. Iffor each Y It+1(G*).then G is Almost-Hamilton-Connected.Theorem 3 Let G be a (fc + l)-connected graph of order n with k > 3.6 be an interger and 0 < b < k. b' = min{k.(2b-1+k)/2}. Iffor each Y (G), then G is Almost-Hamilton-Connected.Theorem 4 Let G be a (k + l)-connected graph of order n with k > 2.b be an interger and 0 < b < k. Iffor each Y I(e)k (G), then G is Hamilton-Connected.