The Graphs \(C_5^{(t)}\) are Graceful for \(t \equiv 0,3 \pmod 4\)

Abstract

Given \(t\geq 2\) cycles \(C_n\) of length \(n \geq 3\), each with a fixed vertex \(v^i_0\), \(i=1,2,\ldots,t\), let \(C^(t)_n\) denote the graph obtained from the union of the \(t\) cycles by identifying the \(t\) fixed vertices (\(v^1_0 = v^2_0 = \cdots = v^t_0\)). Koh et al. conjectured that \(C^(t)^n\) is graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(t = 3, 6, 4k\). In this paper, the conjecture is shown to be true for \(n = 5\).