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$y_1,y_2,\ldots ,y_m$, respectively, where $x_i$ corresponds to
by adding edges $w_iw_{i+r}$, $i = 1, 2, . . . , m,$ where $i + r$ is modulo $m$. It is easy to see that
those cycles connected by $b$-edges and $c$-edges. Label the cycles as $C_1,C_2,C_3, \ldots ,C_{|G|/|a|}$
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% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
by adding edges $w_iw_{i+r}$, $i = 1, 2, . . . , m,$ where $i + r$ is modulo $m$. It is easy to see that
