# Edge-Odd Graceful Graphs Related to Cycles

## Sirirat Singhun, Ratinan Boonklurb, Apinya Tirasuwanwasee, Pensiri Sompong

## Keywords:

edge-odd graceful, the cartesian product of a graph, the union of graphs## 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.

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## How to Cite

*Thai Journal of Mathematics*, 15–26. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/932