Transversals and Matroid Partition*

In sec tion 1, transversal Inatroids are assoc iated with "syste ms of di s tinct represe ntatives" (i. e., tra nsversals) and,. more generally , matc hing matroids are associated with matc hings in graphs . Th e transversal matrolds and a theore m of P. J. Higgins on disjoint transversals of a famil y of se ts , along wIth th e well-known graphIc matrold s a nd some theorems on decomposition of graphs into fore sts motI~ate som e theore ms on partitions of general matroid s into inde pe ndent sets. In sec tion 2, th ~ relatIOns hiP . between transversal result and matroid result is illustrated for a s pecial case of later theorems. In sectIOn 3, theorems on trans versals are proved using network flows . In sec tions 4 and 5, theorems on matrolds are presented which imply various res ults on decomposition into transversals 0,1' IIltO forests . In sec tion 6, the matc hing matro ids are show n to be simply the transversal matroid s. 1' 01' the mos t part , sec tI ons 2, 3, 45, and 6 can be read separately.