Edge-Odd Graceful Graphs Related to Cycles

Sirirat Singhun, Ratinan Boonklurb, Apinya Tirasuwanwasee, Pensiri Sompong

Abstract

Let $G$ be a graph consisting of the vertex set  $V(G)$ and the edge set $E(G)$ such that $|E(G)|=q$. An \emph{edge-odd graceful labeling} is a bijection function $f:E(G)\rightarrow \{1,3,5,\dots ,2q-1\}$ such that for each $v\in V(G)$, $f^*(v)=\sum_{uv\in E(G)}f(uv)$ (mod $2q$) are all distinct. In this article, edge-odd graceful labelings for graphs related to cycles, $(n,1)$-kite and $(n,2)$-kite where $n$ is an integer such that $n\geq 3$, the graph $P_2\cdot nK_1$ where $n$ is a positive integer and the cartesian product $C_n\Box P_3$ and  $C_3\Box P_k$ where $n\geq 3$ and $k\geq 4$ are obtained. Moreover, we show that if a graph  $G$ is edge-odd graceful and  each vertex has odd degree, then the union of even copies of $G$ is edge-odd graceful.