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# GraphesGraphs and Hypercubeshypercubes

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Consider the oriented n-dimensional hypercubes $C_n$.C_n$. • $C_0$: C_0$: one object $X_0$.X_0$. • $C_1$: C_1$: $X_0 \to X_1$.
• $C_2$:C_2$: $$\xymatrix{X_{00} \ar[r] & \ar[d] rightarrow & X_{01} \ar[d] \\downarrow && \downarrow \\X_{10} & \ar[r] rightarrow & X_{11} X_{11}.\end{center}end{array}$$ • $$\xymatrix{X_{000} & \ar[rr] rightarrow & \ar[dd] rightarrow & \ar[dr] && rightarrow & X_{010} && \ar@{-->}[dd] \ar[dr] \downarrow & \searrow & & & \downarrow & \searrow & \\\downarrow & & X_{100} & \ar[rr] rightarrow & \ar[dd] && rightarrow &\rightarrow & X_{110} \ar[dd] \\downarrow & & \downarrow & & \downarrow & & \downarrow \\X_{001} & \ar@{-->}[rr] rightarrow & \ar[dr] && downarrow & \rightarrow & X_{011} && \ar@{-->}[dr] downarrow \\& \searrow & \downarrow & & & \searrow & \downarrow \\& & X_{101} & \ar[rr] && rightarrow & \rightarrow & \rightarrow & X_{111}\end{center}end{array}$$ And so on, inductively over $n \in \mathbb{N}$. mathbb{N}$. Some of the objects (orthey can be $0$.0$. is included in $C_n$? C_n$? (References are also welcomed.)

Example 1: $I = { \xymatrix{X \ar@<+0.35ex>[r]^f { X \ar@<-0.35ex>[r]_g & overset{f}{\underset{g}\rightrightarrows} Y } }$ \}$ can bearranged on $C_2$:C_2$:
$$\xymatrix{ X \ar[r]^f & \ar[d]_g overset{f}\rightarrow & Y \ar[d] \ {\scriptstyle g}\downarrow && \downarrow \\ Y & \ar[r] rightarrow & 0.\end{center}end{array}$$

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# Graphes and Hypercubes

Consider the oriented n-dimensional hypercubes $C_n$.
$C_0$: one object $X_0$.
$C_1$: $X_0 \to X_1$.
$C_2$:
\begin{center} \xymatrix{ X_{00} \ar[r] \ar[d] & X_{01} \ar[d] \ X_{10} \ar[r] & X_{11} . } \end{center}
$C_3$:
\begin{center} \xymatrix{ X_{000} \ar[rr] \ar[dd] \ar[dr] && X_{010} \ar@{-->}[dd] \ar[dr] \ & X_{100} \ar[rr] \ar[dd] && X_{110} \ar[dd] \ X_{001} \ar@{-->}[rr] \ar[dr] && X_{011} \ar@{-->}[dr] \ & X_{101} \ar[rr] && X_{111} . } \end{center}
And so on, inductively over $n \in \mathbb{N}$. Some of the objects (or everyone) filling the vertices can eventually be the same. Moreover, they can be $0$.

Let $I$ be an arbitrary finite oriented diagram (graph) with no cycles (composable arrows starting and ending on one single object).

My question is: Does always exist an $n \in \mathbb{N}$ such that $I$ is included in $C_n$? (References are also welcomed.)

Example 1:
$I = { \xymatrix{X \ar@<+0.35ex>[r]^f \ar@<-0.35ex>[r]_g & Y} }$ can be arranged on $C_2$:
\begin{center} \xymatrix{ X \ar[r]^f \ar[d]_g & Y \ar[d] \ Y \ar[r] & 0 . } \end{center}