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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 ≥ 1 , let σ t ( G ) be the smallest sum of degrees of t independent vertices of G . We prove that if G has order at least 7 k + 1 and σ 4 ( G ) ≥ 8 k − 3 , with k ≥ 2 , then G contains k vertex-disjoint cycles. We also show that...
The neighborhood degree list (NDL) is a graph invariant that refines information given by the degree sequence and joint degree matrix of a graph and is useful in distinguishing graphs having the same degree sequence. We show that the space of realizations of an NDL is connected via a switching operation. We then determine the NDLs that have a unique realization by a labeled graph; the characterization...
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