Analogy of Fan-type Condition on Weak Cycle Partition of Graphs

For a graph G of order n and a positive integer k, a k-weak cycle partition of G, called k-WCP, is a sequence of vertex disjoint subgraphs H₁, H₂, ⋯, H_k of G with \(\bigcup\nolimits_{i=1}^{k} V(H_{i})=V(G)\), where H_i is isomorphic to K₁, K₂ or a cycle. Let σ₂(G) = min{d(x) + d(y): xy ∉ E(G), x, y ∈ V(G)}. Hu and Li [Discrete Math. 307(2007)] proved that if G is a graph of order n ≥ k + 12 with a k-WCP and \(\sigma_{2}(G) \geq {{2n+k-4} \over 3}\), then G contains a k-WCP with at most one subgraph isomorphic to K₂. In this paper, we generalize their result on the analogy of Fan-type condition that \(\max\{{d(x),d(y)}\} \geq {{2n+k-4} \over 6}\) for each pair of nonadjacent vertices x, y ∈ V(G).