The Graphs \(C_{13}^{(t)}\) are Graceful for \(t \equiv 0,3 \pmod{4}\)

Abstract

Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0,3 \pmod{4}\). The conjecture has been shown true for \(n = 3,5,6,7,9,11,4k\). In this paper, the conjecture is shown to be true for \(n = 13\).