Edge odd graceful labeling of some path and cycle related graphs

Peer review under responsibility of Kalasalingam University.

Figures & data

Fig. 1 Friendship graph $F{r}_n^{\left(3\right)}$, $n\equiv 0\left(mod3\right)$.

Fig. 2 Friendship graph $F{r}_n^{\left(3\right)}$, $n\equiv 1\left(mod3\right)$.

Fig. 3 Friendship graph $F{r}_n^{\left(3\right)}$, $n\equiv 2\left(mod3\right)$.

Fig. 4 The graphs $F{r}_9^{\left(3\right)}$, $F{r}_{10}^{\left(3\right)}$ and $F{r}_{11}^{\left(3\right)}$ are edge odd graceful graphs.

Fig. 5 Friendship graph $F{r}_n^{\left(4\right)}$, $n$ is odd.

Fig. 6 Friendship graph $F{r}_n^{\left(4\right)}$, $n$ is even.

Fig. 7 The graphs $F{r}_9^{\left(4\right)}$ and $F{r}_{10}^{\left(4\right)}$ are edge odd graceful graphs.

Fig. 8 Friendship graph $\bar{F}{r}_n^{\left(3\right)}$.

Fig. 9 The graphs $\bar{F}{r}_9^{\left(3\right)}$, $\bar{F}{r}_{10}^{\left(3\right)}$, $\bar{F}{r}_{11}^{\left(3\right)}$ and $\bar{F}{r}_{25}^{\left(3\right)}$ are edge odd graceful graphs.

Fig. 10 Wheel graph $W_n$, $n$ is even.

Fig. 11 Wheel graph $W_n$, $n\equiv 1\left(mod4\right)$.

Fig. 12 Wheel graph $W_n$, $n\equiv 3\left(mod4\right)$.

Fig. 13 The graphs $W_8$, $W_9$, $W_{10}$ and $W_{11}$ are edge odd graceful graphs.

Fig. 14 Helm graph $H_n$, $n$ is odd.

Fig. 15 Helm graph $H_n$, $n$ is even.

Fig. 16 The graphs $H_9$, and $H_{10}$ are edge odd graceful graphs.

Fig. 17 Web graph $W{b}_n$.

Fig. 18 The graphs $W{b}_{11}$, and $W{b}_{12}$ are edge odd graceful labeling graphs.

Fig. 19 Double wheel graph $W_{n,n}$.

Fig. 20 The graphs $W_{8,8}$, and $W_{9,9}$ are edge odd graceful labeling graphs.

Fig. 21 Fan graph $F_n$, $n\equiv 0\left(mod4\right)$ or $n\equiv 3\left(mod4\right)$.

Fig. 22 Fan graph $F_n$, $n\equiv 1\left(mod4\right)$ or $n\equiv 2\left(mod4\right)$.

Fig. 23 The graphs $F_8$, $F_9$, $F_{10}$ and $F_{11}$ are edge odd graceful graphs.

Fig. 24 Gear graph $G_n$.

Fig. 25 The graphs $G_6$, $G_7$, $G_8$, $G_9$, $G_{10}$ and $G_{11}$ are edge odd graceful graphs.

Fig. 26 Half gear graph $H{G}_n$, $n\equiv 0\left(mod6\right)$ or $n\equiv 3\left(mod6\right)$.

Fig. 27 Half gear graph $H{G}_n$, $n\equiv 1\left(mod6\right)$ or $n\equiv 5\left(mod6\right)$.

Fig. 28 Half gear graph $H{G}_n$, $n\equiv 2\left(mod6\right)$.

Fig. 29 Half gear graph $H{G}_n$, $n\equiv 9\left(mod6\right)$.

Fig. 30 $H{G}_6$, $H{G}_7$, $H{G}_8$, $H{G}_9$, $H{G}_{10}$ and $H{G}_{11}$ are edge odd graceful labeling graphs.

Fig. 31 Double fan graph $F_{2,n}$, $n$ is odd or $n\equiv 2\left(mod12\right)$ or $n\equiv 4\left(mod12\right)$ or $n\equiv 6\left(mod12\right)$.

Fig. 32 Double fan graph $F_{2,n}$, $n\equiv 0\left(mod12\right)$ or $n\equiv 8\left(mod12\right)$ or $n\equiv 10\left(mod12\right)$.

Fig. 33 The graphs $F_{2,6}$, $F_{2,7}$, $F_{2,8}$, $F_{2,9}$, $F_{2,10}$, $F_{2,11}$, $F_{2,12}$, $F_{2,14}$ and $F_{2,16}$ are edge odd graceful graphs.

Fig. 34 Polar grid graph $P_{m,n}$, $m$ is even.

Fig. 35 Polar grid graph $P_{m,n}$, $m$ is odd.

Fig. 36 The graphs $P_{4,15}$, $P_{5,13}$, $P_{5,16}$, $P_{6,12}$ and $P_{7,11}$ are edge odd graceful graphs.