Simple groups and strong covers

Given a finite, nonabelian, simple group S and labelling π as the set of prime divisors of |S|, a set C of character degrees of S strongly covers π if some fixed prime divides every member of C and every prime in π divides at least one member of C. In this article, we classify the simple groups S for which there is such a set.When there is such a set, we associate a strong covering number scn(S) with S by letting this be the cardinality of a smallest set C of degrees which strongly covers π. Finding a set C which strongly covers π establishes an upper bound on scn(S), but in many cases our set C has minimal cardinality, and so scn(S)=|C|. In addition, if cd(S) has a subset C which strongly covers π, C can be chosen so |C|⩽3, showing scn(S)⩽3, and we furthermore classify which simple groups have a strong covering number of 3. For all the sporadic and alternating groups, and several families of groups of Lie type, we calculate the exact strong covering number. Several consequences of these facts are also presented.