Degree sums and path-factors in graphs (Q5936092)

A spanning subgraph \(H\) of a graph \(G\) is called a path-factor if each component of \(H\) is a path of order at least 2. Denote by \(\sigma_3(G)\) the smallest of the sums of degrees of triples of vertices that form independent sets in \(G\). Let \(n_1,n_2,\dots, n_k\) be a sequence of integers larger than 1 such that \(n_1+ n_2+\cdots+ n_k= n\) and let \(\lambda\) be the number of even integers in this sequence. The authors prove that if \(G\) is a connected graph of order \(n\) and \(\sigma_3(G)\geq{3\over 2}(n- k+\lambda)\) then \(G\) contains a path-factor consisting of paths of orders \(n_1,n_2,\dots, n_k\). This theorem is a generalization of an earlier result by R. Johansson who proved a similar statement with the assumption \(\sigma_3(G)\geq {3\over 2}(n- k+\lambda)\) replaced by the stronger condition \(\delta(G)\geq{1\over 2}(n- k+\lambda)\). The authors argue that their result is in some sense best possible.