On the Minimal Density of Triangles in Graphs
For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that $$ g_3(\rho) =\frac{(t-1)\ofb{t-2\sqrt{t(t-\rho(t+1))}}\ofb{t+\sqrt{t(t-\rho(t+1))}}^2}{t^2(t+1)^2},$$ where $t\df \lfloor...
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| Veröffentlicht in: | Combinatorics, probability & computing probability & computing, 2008-07, Vol.17 (4), p.603-618 |
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| container_title | Combinatorics, probability & computing |
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| creator | RAZBOROV, ALEXANDER A. |
| description | For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that
$$ g_3(\rho)
=\frac{(t-1)\ofb{t-2\sqrt{t(t-\rho(t+1))}}\ofb{t+\sqrt{t(t-\rho(t+1))}}^2}{t^2(t+1)^2},$$
where $t\df \lfloor 1/(1-\rho)\rfloor$ is the integer such that $\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$. |
| doi_str_mv | 10.1017/S0963548308009085 |
| format | Article |
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$$ g_3(\rho)
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$$ g_3(\rho)
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$$ g_3(\rho)
=\frac{(t-1)\ofb{t-2\sqrt{t(t-\rho(t+1))}}\ofb{t+\sqrt{t(t-\rho(t+1))}}^2}{t^2(t+1)^2},$$
where $t\df \lfloor 1/(1-\rho)\rfloor$ is the integer such that $\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.1017/S0963548308009085</doi><tpages>16</tpages></addata></record> |
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| source | Cambridge University Press Journals Complete |
| title | On the Minimal Density of Triangles in Graphs |
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