The Shields-Harary numbers of \(K_{m,n}\) for continuous concave cost functions vanishing at 1 (Q2462353)

Given a finite simple graph \(G=(V,E)\) and a weighting \(g:V\to[0,\infty)\) of its vertices. For an non-increasing cost function \(f:[0,\infty)\to [0,\infty]\), set \(\overline{m}_f(g,G)\) to be the minimum of \(\sum_{u\in S}f(g(u))\) over all subsets \(S\) of the set of vertices \(V\) such that \(\sum_{v\in V(H)}g(v)< 1\) holds for all components of \(G-S\). Then the Shields-Harary number \(\overline{SH}(G,f)\) of \(G\) with respect to \(f\) is the supremum of \(\overline{m}_f(g,G)\) over all weightings \(g:V\to[0,\infty)\). The authors show that for \(f=1-x\) and the complete bipartite graph \(G=K_{m,n}\), the Shields-Harary number \(\overline{SH}(G,f)\) equals \(m\), if \(n\geq m+2\sqrt{m}\), and equals \({1\over n+1}\lfloor(n+m)^2/4\rfloor\), if \(m\leq n<m+2\sqrt{m}\). It follows that the Shields-Harary number of \(K_{m,n}\) with respect to any concave continuous cost function \(f\) on \([0,1]\) satisfying \(f(1)=0\) is \(mf(0)\), if \(n\geq m+2\sqrt{m}\), and between \({1\over n+1}\lfloor(n+m)^2/4\rfloor f(0)\) and \(mf(0)\), if \(m\leq n<m+2\sqrt{m}\).