Turán number for \(pS_r\) (Q2829351)

The Turán number \(\operatorname{ex}(m,G)\) of a graph \(G\) is the maximal number of edges in any simple graph with \(m\) vertices having no subgraph isomorphic to \(G\).NEWLINENEWLINEThe authors of the present paper prove that if \(G = pS_r\) is the disjoint union of \(p\) copies of the star graph \(S_r = K_{1,r}\), then NEWLINE\[NEWLINE \operatorname{ex}(m,pS_r) = \lfloor (m-p+1)(r-1)/2 \rfloor + (p-1)m - \binom{p}{2} NEWLINE\]NEWLINE for all \(p,r\geq 1\) and \(m\geq r^2 \binom{p}{2} + p-2 + \max\{rp,r^2+2r\}\).NEWLINENEWLINEThis improves a special case of a previously proved expression, for \(m\) sufficiently large, for the Turán number of disjoint unions of stars not necessarily of the same size; see [\textit{B. Lidicky} et al., Electron. J. Comb. 20, No. 2, Research Paper P62, 13 p. (2013; Zbl 1298.05071)].