科研之友
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摘要
A graph G is s-Hamiltonian if for any S subset of V(G) of order at most s, G - S has a Hamiltonian-cycle, and s-Hamiltonian connected if for any S subset of V(G) of order at most s, G - S is Hamiltonian-connected. Let k > 0, s >= 0 be two integers. The following are proved in this paper: (1) Let k >= s + 2 and s <= n - 3. If G is a k-connected graph of order n and if max {d(v) : v is an element of I} >= (n+s)/2 for every independent set I of order k-s such that I has two distinct vertices x, y with 1 <= vertical bar N(x) boolean AND N(y)vertical bar <= alpha(G) + s -1, then G is s-Hamiltonian. (2) Let k >= s + 3 and s <= n - 2. If G is a k-connected graph of order n and if max{d(v) : v is an element of I} >= (n + s + 1)/2 for every independent set I of order k - s - 1 such that I has two distinct vertices x, y with 1 <= vertical bar N(x) boolean AND N(y)vertical bar <= alpha(G) + s, then G is s-Hamiltonian connected. These extended several former results by Dirac, Ore, Fan and Chen.
