On ternary transversal matroids

Ingleton [8, p. 123] raised the question of characterizing the class of transversal matroids that are representable over some particular field F. He noted that when F = GF(2), this problem had already been solved by de Sousa and Welsh [6] who showed that the class of binary transversal matroids coincides with the class of graphic transversal matroids. The latter class had earlier been characterized by Bondy [2] and Las Vergnas [10]. Let C 2 be the graph that is obtained from a k-edge cycle by adding a new edge in parallel to each existing edge. On combining the above-mentioned results with Crapo and Rota's Scum Theorem [15, p. 324] and Tutte's well-known characterization of binary matroids [15, p. 167], we get that a matroid is binary and transversal if and only if it has no series minor isomorphic to U2,4, M(K4) or M(C~) for any k >~ 3. The purpose of this paper is to give a similar excluded-series-minor characterization of the class of ternary transversal matroids and thereby answer Ingleton's question when f = GF(3). The matroid terminology used here will in general follow Welsh [15]. If S is a set, then S = X l t . J X 2 U . . . U X,~ indicates that S is the disjoint union of X1, X2, • • • , Xm. The ground set and rank of the matroid M will be denoted by E(M) and rk M respectively. If T ~_ E(M) , then the rank of T will be written as rk T and we shall denote the deletion of T from M by M \ T or M I ( E ( M ) T). The contraction of T from M will be denoted M / T . We shall sometimes write N ~_ M to indicate that N is a restriction of M having the same rank as M. Flats of M of ranks one and two will be called points and lines. A line is non-trivial if it contains at least three points. If M1 and M2 are matroids on the sets S and S t3 e, then M2 is an extension of M1 if M2\e = 1t41, and M2 is a lift of M1 if M~ is an extension of M~'. We call M 2 a non-trivial extension of M~ if e is neither a loop nor a coloop of M2 and e is not in