Abstract
For a graph G, we define σ2(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v 1,...,v k , G has k vertex-disjoint cycles C 1,..., C k of length at most four such that v i ∈ V(C i ) for all 1 ≤ i ≤ k. And show if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v 1,...,v k , G has k vertex-disjoint cycles C 1,..., C k such that v i ∈ V(C i ) for all 1 ≤ i ≤ k, V(C 1) ∪...∪ V(C k ) = V(G), and |C i | ≤ 4 for all 1 ≤ i ≤ k − 1.
The condition of degree sum σ2(G) ≥ n + k − 1 is sharp.
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Received: December 20, 2006. Final version received: December 12, 2007.
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Dong, J. A 2-factor with Short Cycles Passing Through Specified Independent Vertices in Graph. Graphs and Combinatorics 24, 71–80 (2008). https://doi.org/10.1007/s00373-008-0776-x
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DOI: https://doi.org/10.1007/s00373-008-0776-x