Conditional Edge Connectivity of the Locally Twisted Cubes

The k-component edge connectivity \(c\lambda _{k}(G)\) of a non-complete graph G is the minimum number of edges whose deletion results in a graph with at least k components. In this paper, we extend some results by Guo et al. (Appl Math Comput 334:401–406, 2018) by determining the component edge connectivity of the locally twisted cubes \(\mathrm{LTQ}_{n}\), i.e., \(c\lambda _{k+1}(\mathrm{LTQ}_{n})=kn-\frac{ex_{k}}{2}\) for \(1\leqslant k\leqslant 2^{[\frac{n}{2}]}\), \(n\geqslant 7\), where \(ex_{k}=\sum _{i=0}^{s}t_{i}2^{t_{i}}+\sum _{i=0}^{s}2\cdot i\cdot 2^{t_{i}}\), and k is a positive integer with decomposition \(k=\sum _{i=0}^{s}2^{t_{i}}\) such that \(t_{0}=\lfloor \mathrm{log}_{2}k\rfloor \) and \(t_{i}=\lfloor \mathrm{log}_{2}(k-\sum _{r=0}^{i-1}2^{t_{r}})\rfloor \) for \(i\geqslant 1\). As a by-product, we characterize the corresponding optimal solutions.