On path-sequential labellings of cycles
Consider a labelling of the vertices of the cycle \(C_n\) by the integers \(0, 1, \ldots, n-1\), each vertex obtaining a distinct label. Such a labelling is called \(k\)-sequential, when the \(n\) sums of \(k\) adjacent labels form a set of consecutive integers. Vanderkam has conjectured that there is a \(k\)-sequential labelling of \(C_n\), if and only if \(n\) is odd, or \(k\) is odd. This paper shows that a \(k\)-sequential labelling of \(C_{mn}\) can be obtained from a \(k\)-sequential labelling of \(C_m\). This reduces the number of cases to check the conjecture considerably.
scientific article; zbMATH DE number 1439821