Degeneracy in a transport problem arises when the number of occupied cells is less then:

(number of columns + number of rows)-1

the phenomenon of obtaining a degenerate basic feasible solution in a linear programming problem known as degeneracy.

electron degeneracy pressure

The degeneracy theory was proposed by George G. Simpson in 1944. Simpson suggested that evolution could lead to the retention of non-functional traits or "degenerate" characteristics in a species.

This is a guess, but I suspect the person means the electron degeneracy pressure.