The Graphs $C_{11}^{(t)}$ are Graceful for $t\equiv 0,1(\mathrm{mod}4)$

Abstract

Let $C_n^{(t)}$ 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 $nt\equiv 0,3(\mathrm{mod}4)$. The conjecture has been shown true for $n=3,5,6,7,9,4k$. In this paper, the conjecture is shown to be true for $n=11$.