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Example 1. Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.

Example 2. Let $n=2$, $m=15$. Then $c=5$, as $G=3K_5$ is a $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z_{15}$ has chromatic number $3$.

Example 1. Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.

Example 2. Let $n=2$, $m=15$. Then $c=5$, as $G=3K_5$ is $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z_{15}$ has chromatic number $3$.

Example 1. Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.

Example 2. Let $n=2$, $m=15$. Then $c=5$, as $G=3K_5$ is a $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z_{15}$ has chromatic number $3$.

Source Link
bof
bof
  • 13.4k
  • 2
  • 43
  • 66

Example 1. Let $n=1$, $m=7$. Then $c=3$, and $G=C_3+C_4$ is a $2$-regular graph of order $7$ and chromatic number $3$, but is not isomorphic to $\mathbb Z_7$.

Example 2. Let $n=2$, $m=15$. Then $c=5$, as $G=3K_5$ is $4$-regular graph of order $15$ and chromatic number $5$, but $\mathbb Z_{15}$ has chromatic number $3$.