The toroidal splitting number of the complete graph \(K_ n\)
The toroidal splitting number \(\sigma (G,S_ 1)\) is the smallest number k such that G can be obtained from a graph imbeddable in the torus by k vertex identifications. The author establishes that \(\sigma (K_ n,S_ 1)=\lceil (n-3)(n-4)/6\rceil -2\) for \(n\geq 8\); \(\sigma (K_ n,S_ 1)=0\) otherwise. (It is worth noting that, for \(n\geq 8\), the value is exactly two less than the nonorientable genus for the same graphs.) The proof uses Euler considerations for the lower bound, current graphs to obtain duals of splittings of \(K_{12s+1}\) and \(K_{12s+7}\), and modifications of these duals for the other ten cases. Splitting numbers are related to the pseudocharacteristic of a graph; see, for example, Section 6-7 of the reviewer's book ''Graphs, groups and surfaces (revised edition 1984; Zbl 0551.05037) [see also the review of the first edition (1973) in Zbl 0268.05102.] Thus the present constructions show that, for \(n\geq 8\), the pseudocharacteristic of \(K_ n\) is at least -\(\lceil n(n-7)/6\rceil\). An easy Euler argument shows that this is best possible - that is, \(K_ n\) has pseudocharacteristic -\(\lceil n(n-7)/6\rceil\), for \(n\geq 8\). As far as this reviewer knows, this has not been observed before. (The pseudocharacteristics of \(K_ 3\), \(K_ 4\), \(K_ 5\), and \(K_ 7\) also fit the formula; the pseudocharacteristics of \(K_ 1\) and \(K_ 2\) are trivially 2, and for \(K_ 6\) it is less trivially 0.)
scientific article; zbMATH DE number 3963857
scientific article