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For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero.

For general $t$, your claim is that if $G$ is the circulant graph of order $n$ with connection set $S$ and the sum of the elements in $S$ is nonzero modulo $n$, then there is a simple path in $G$ using every element of $S$ exactly once. I think that Brian Alspach conjectured this to be true, but I can't find a reference at the moment.

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero.

For general $t$, your claim is that if $G$ is the circulant graph of order $n$ with connection set $S$ and the sum of the elements in $S$ is nonzero modulo $n$, then there is a simple path in $G$ using every element of $S$ exactly once. I think that this is an open problem.

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero.

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero.

For general $t$, your claim is that if $G$ is the circulant graph of order $n$ with connection set $S$ and the sum of the elements in $S$ is nonzero modulo $n$, then there is a simple path in $G$ using every element of $S$ exactly once. I think that Brian Alspach conjectured this to be true, but I can't find a reference at the moment.

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero. Now if you drop this assumption, can you always find a simple cycle that uses every length exactly once?

For $t=n-1$ you are asking if the cyclic group of order $n$ is sequenceable, where a finite group $G$ is sequenceable if you can write down the non-identity elements $g_1,\ldots,g_{n-1}$ in such a way that the products $e,g_1,\,g_1g_2,\ldots,g_1g_2\cdots g_{n-1}$ are distinct. Gordon (Sequences in groups with distinct partial products, Pacific J. Math, 1961 ) proved that this is the case if and only if $n$ is even, which in your case follows from the assumption that the sum of the $l_i$ is nonzero.