For a graph G, define \(\sigma _2(G)\) by \(\sigma _2(G)=\min \bigl \{d_G(x)+d_G(y):x, y\in V(G), x\ne y, xy\notin E(G)\bigr \}\). If G is a bipartite graph with partite sets X and Y, we also define \(\sigma _{1,1}(G)\) by \(\sigma _{1,1}(G)=\min \{d_G(x)+d_G(y):x\in X, y\in Y, xy\notin E(G)\}\). Ore’s theorem states that a graph of order \(n\ge 3\) with \(\sigma _2(G)\ge n\) contains a hamiltonian cycle and the Moon–Moser theorem states that a balanced bipartite graph G of order \(2n\ge 4\) with \(\sigma _{1,1}(G)\ge n+1\) contains a hamiltonian cycle. In Chen et al. (Discrete Math 343:Article No. 111663, 2020), we studied the relationship between Ore’s theorem and the Moon–Moser theorem, and proved that the refinement of the Moon–Moser theorem given by Ferrara et al. (Discrete Math 312:459–461, 2012) implies Ore’s theorem for graphs of even order. In this paper, we extend the above study to the graphs of odd order. Since no graphs of odd order contain a spanning balanced bipartite subgraph, the Moon–Moser theorem does not work in this case. We instead introduce its counterpart for the graphs in which the orders of the partite sets differ by 1, proved in Matsubara et al. (Discrete Math 340:87–95, 2017). We refine this result and prove that this refinement implies Ore’s theorem.

对于图G，定义\(\sigma _2(G)\)由\(\sigma _2(G)=\min \bigl \{d_G(x)+d_G(y):x, y\in V(G) ), x\ne y, xy\notin E(G)\bigr \}\)。如果G是具有分集X和Y的二分图，我们还定义了\(\sigma _{1,1}(G)\)由\(\sigma _{1,1}(G)=\min \{ d_G(x)+d_G(y):x\in X, y\in Y, xy\notin E(G)\}\)。矿石定理指出的顺序的图\（N \ GE 3 \）与\（\西格玛_2（G）\ GEÑ\）含有哈密顿周期和月亮-Moser的定理指出均衡二分图G ^的顺序\ (2n\ge 4\)与\(\sigma _{1,1}(G)\ge n+1\)包含一个哈密顿循环。在陈等人。(Discrete Math 343:Article No. 111663, 2020)，我们研究了Ore定理和Moon-Moser定理的关系，证明了Ferrara等人给出的Moon-Moser定理的细化。(Discrete Math 312:459–461, 2012) 暗示了偶数阶图的 Ore 定理。在本文中，我们将上述研究扩展到奇数阶图。由于没有奇数阶图包含跨越平衡二部子图，因此 Moon-Moser 定理在这种情况下不起作用。相反，我们为部分集的阶数相差 1 的图引入了它的对应物，这在 Matsubara 等人中得到了证明。（离散数学 340：87-95，2017 年）。我们改进这个结果并证明这个改进意味着 Ore 定理。