\(K_p\)-Removable Sequences of Graphs

Let \(\{G_{pn} | n \geq 1\} = \{G_{p1}, G_{p2}, G_{p3}, \ldots\}\) be a countable sequence of simple graphs, where \(G_{pn}\) has \(pn\) vertices. This sequence is called \(K_p\)-removable if \(G_{p1} = K_p\), and \(G_{pn} – K_p = G_{p(n-1)}\) for every \(n \geq 2\) and for every \(K_p\) in \(G_{pn}\). We give a general construction of such sequences. We specialize to sequences in which each \(G_{pn}\) is regular; these are called regular \((K_p, \lambda)\)-removable sequences, where \(\lambda$ is a fixed number, \(0 \leq \lambda \leq p\), referring to the fact that \(G_{pn}\) is \((\lambda(n – 1) + p – 1)\)-regular. We classify regular \((K_p, 0)\)-, \((K_p, p – 1)\)-, and \((K_p, p)\)-removable sequences as the sequences \(\{nK_p | n \geq 1\}\), \(\{K_{p \times n} | n \geq 1\}\), and \(\{K_{pn} | n \geq 1\}\) respectively. Regular sequences are also constructed using `levelled’ Cayley graphs, based on a finite group. Some examples are given.