# Do we really need degree constraints for ILP formulations of TSP problems

The Dantzig-Fulkerson ILP-formulation of the symmetric TSP is $$\min\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^n c_{ij}x_{\lbrace i,j\rbrace}\quad\text{s.t.}\\ \sum\limits_{j\ne i,\,j=1}^n x_{\lbrace i,j\rbrace}=2\quad i=1,\,\dots,\, n;\\ \sum\limits_{i\in Q}\sum\limits_{j\ne i,\,j\in Q}x_{\lbrace i,j\rbrace}\le|Q|-1\quad \forall Q\subsetneq \lbrace1,\,\dots,\,n\rbrace,\ |Q|\ge 2\\ x_{\lbrace i,j\rbrace}\in\lbrace 0,1\rbrace$$

Questions:

• is it known that the $$n$$ degree-constraints for vertices be replaced with the single cardinality constraint $$\sum\limits_{i=1}^{n-1}\sum_{j=i+1}^n x_{\lbrace i,j\rbrace}=n$$ because the maximal number of edges that a cycle-free graph with $$n$$ vertices can contain is $$n-1$$?
• what are the tradeoffs for the reduced number of constraints?