Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known, where path length is measured by Euclidean distance in $\mathbb{R}^d$? The unit hypercube spans from $(0,0,\ldots,0)$ to $(1,1,\ldots,1)$.

For example, for the cube in $\mathbb{R}^3$, I believe the longest path has length $3\sqrt{2}+4\sqrt{3} \approx 11.17$, avoiding all edges of length $1$, and using all $3$ of the long diagonals:

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          [![LongestCubePath][1]][1]
          ^{Path: $(1,7,2,8,3,5,4,6)$.}

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This likely has been studied, in which case pointers would be appreciated. If exact values are not known, bounds would be useful.

Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known, where path length is measured by Euclidean distance in $\mathbb{R}^d$? The unit hypercube spans from $(0,0,\ldots,0)$ to $(1,1,\ldots,1)$.

For example, for the cube in $\mathbb{R}^3$, I believe the longest path has length $3\sqrt{2}+4\sqrt{3} \approx 11.17$, avoiding all edges of length $1$, and using all $4$ of the long diagonals and $3$ short diagonals:

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          [![LongestCubePath][1]][1]
          ^{Path: $(1,7,2,8,3,5,4,6)$.}

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This likely has been studied, in which case pointers would be appreciated. If exact values are not known, bounds would be useful.

Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known, where path length is measured by Euclidean distance in $\mathbb{R}^d$? The unit hypercube spans from $(0,0,\ldots,0)$ to $(1,1,\ldots,1)$.

For example, for the cube in $\mathbb{R}^3$, I believe the longest path has length $3\sqrt{2}+4\sqrt{3} \approx 11.17$, avoiding all edges of length $1$, and using all $3$ of the long diagonals:

---

          [![LongestCubePath][1]][1]
          ^{Path: $(1,7,2,8,3,5,4,6)$.}

---

This likely has been studied, in which case pointers would be appreciated. If exact answers are not known, bounds would be useful.

Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known, where path length is measured by Euclidean distance in $\mathbb{R}^d$? The unit hypercube spans from $(0,0,\ldots,0)$ to $(1,1,\ldots,1)$.

For example, for the cube in $\mathbb{R}^3$, I believe the longest path has length $3\sqrt{2}+4\sqrt{3} \approx 11.17$, avoiding all edges of length $1$, and using all $3$ of the long diagonals:

---

          [![LongestCubePath][1]][1]
          ^{Path: $(1,7,2,8,3,5,4,6)$.}

---

This likely has been studied, in which case pointers would be appreciated. If exact values are not known, bounds would be useful.