I'm reading this articel which is about formulating VCP to eleminate symmetric solution, they say:
And then In order to eliminate some of the symmetrical solutions, they say these two constraint is enough :
My question is: Why do we need to have equation 5. I've tried to compare different symmetric coloring and it seems that only having equation 6 is enough to ensure asymtric solution. Right? So, why we have to have 5 as well?
Constraint $(5)$ enforces the logical implication $w_i = 1 \implies \bigvee_{v\in V} (x_{vi} = 1)$. Without it, you could have $w_i=1$ but $x_{vi}=0$ for all $v$, which means that you didn't really use color $i$. At optimality, this would not happen because you could improve the objective by permuting the colors, but imposing $(5)$ ensures that every feasible solution uses only consecutive colors starting from $1$.