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\iteman{io-port 05823676}
\itemau{Akbari, S.; Jamaali, M.; Mahmoody, A.; Fakhari, Seyed S.A.}
\itemti{On the size of graphs whose cycles have length divisible by a fixed integer.}
\itemso{Australas. J. Comb. 45, 67-72 (2009).}
\itemab
Summary: Let $G$ be a simple graph of order $n$ and size $m$ which is not a tree. If $\ell\ge 3$ is a natural number and the length of every cycle of $G$ is divisible by $\ell$, then $m\le{\ell\over\ell- 2}(n-2)$, and the equality holds if and only if the following hold: (i) $\ell$ is odd and $G$ is a cycle of order $f$ or (ii) $\ell$ is even and $G$ is a generalized $\theta$-graph with paths of length 2. It is shown that for a $(0\text{\,mod\,} \ell)$-cycle graph, ${m\over n}< {\ell\over\ell-1}$, if $\ell$, is odd, and for a given $\varepsilon> 0$, there exists a $(0\text{\,mod\,}\ell)$-cycle graph $G$ with ${m\over n}> {\ell\over\ell- 1}-\varepsilon$. Also ${m\over n}<{\ell\over\ell- 2}$, if $\ell$ is even, and for a given $\varepsilon> 0$, there exists a $(0\text{\,mod\,}\ell)$-cycle graph $G$ with ${m\over n}> {\ell\over\ell- 2}-\varepsilon$.
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