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

Abstract

Given $t\ge 2$ cycles $C_n$ of length $n\ge 3$, each with a fixed vertex $v_0^i$, $i=1,2,\dots ,t$, let $C^{(}t{)}_n$ denote the graph obtained from the union of the $t$ cycles by identifying the $t$ fixed vertices ($v_0^1=v_0^2=\cdots =v_0^t$). Koh et al. conjectured that $C^{(}t{)}^n$ is graceful if and only if $nt\equiv 0,3(\mathrm{mod}4)$. The conjecture has been shown true for $t=3,6,4k$. In this paper, the conjecture is shown to be true for $n=5$.