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On vertex-disjoint cycles and degree sum conditions. (English) Zbl 1372.05111
Summary: This paper considers a degree sum condition sufficient to imply the existence of $$k$$ vertex-disjoint cycles in a graph $$G$$. For an integer $$t$$, let $$\sigma_t(G)$$ be the smallest sum of degrees of t independent vertices of $$G$$. We prove that if $$G$$ has order at least $$7k+1$$ and $$\sigma_4(G)\geq 8k-3$$, with $$k$$, then $$G$$ contains $$k$$ vertex-disjoint cycles. We also show that the degree sum condition on $$\sigma_4(G)$$ is sharp and conjecture a degree sum condition on $$\sigma_t(G)$$ sufficient to imply $$G$$ contains $$k$$ vertex-disjoint cycles for $$k\geq 2$$.

##### MSC:
 05C38 Paths and cycles 05C07 Vertex degrees 05C35 Extremal problems in graph theory
##### Keywords:
vertex-disjoint cycles; minimum degree sum; degree sequence
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##### References:
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