A New Idea For Hamiltonian Problem

Let \(G\) be a \(2\)-connected graph of order \(n\) with connectivity \(\kappa\) and independence number \(\alpha\). In this paper, we show that if for each independent set \(S\) with \(|S| = k+1\), there are \(u,v \in S\) such that satisfying one of the following conditions:

1. \(d(u) + d(v) \geq n\); or \(|N(u) \cap N(v)| \geq \alpha\); or \(|N(u) \cup N(v)| \geq n-k\);
2. for any \(x \in \{u,v\}\), \(y \in V(G)\) and \(d(x,y) = 2\), it implies that \(\max\{d(x), d(y)\} \geq n/2\),

then \(G\) is hamiltonian. This result reveals the internal relations among several well-known sufficient conditions: \((1)\) it shows that it does not need to consider all pairs of nonadjacent or distance two vertices in \(G\); \((2)\) it makes known that for different pairs of vertices in \(G\), it permits to satisfy different conditions.