Maximum Neighbouring Sums

The vertices of a regular 10-gon are labeled \(V_1, V_2, \ldots V_n\), which is a permutation of \( \{ 1, 2, \ldots, 10\}\). Define a neighboring sum to be the sum of 3 consecutive vertices \( V_i, V_{i+1}\) and \(V_{i+2}\) [where \(V_{11}=V_1, V_{12}=V_2\)]. For each permutation \(\sigma\), let \(N_\sigma\) denote the maximum neighboring sum. As \(\sigma\) ranges over all permutations, what is the minimum value of \(N_\sigma\)?

Details and assumptions

If the integers are written as \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\) around the circle, then the neighboring sums are \(6, 9, 12, 15,\) \(18, 21, 24,\) \(27, 20, 13\), and the maximum neighboring sum is 27.