Representability of Graphical Symplectic Matroids

Abstract

Chow associated with every finite multigraph $G$ a symplectic matroid $\GraphS{G}$ on the signed edge set $E\sqcup E^{*}$, whose restrictions to transversals are frame matroids of signed graphs, and Sarvepalli asked which of these matroids are representable. For a connected graph containing a cycle, we show that every isotropic representation of $S(G)$ admits an incidence-flow normal form. The remaining representation conditions are expressed through its admissible minors. Consequently, if $\GraphS{G}$ is $\F$-representable, then the bridgeless part of $G$ has a nowhere-zero $(\F,+)$-flow. In characteristic other than two, $\GraphS{G}$ is representable if and only if every non-loop edge on a cycle has a parallel mate. In characteristic two, we characterize the loopless graphs without two vertex-disjoint cycles by the existence of a nowhere-zero $(\F,+)$-flow, and we show that rigidly linked short cycles, and hence subdivisions of the prism, are obstructions. For $3$-connected simple graphs this gives a complete answer: $\GraphS{G}$ is $\F$-representable if and only if $\F$ has characteristic two, $G$ has no two vertex-disjoint cycles, and either $|\F|>2$ or $G\cong K_5$. In particular, the matroid of the Petersen graph is representable over no field. Finally, the admissible corank--nullity polynomial of an arbitrary symplectic matroid is intrinsic and satisfies a three-term contraction identity. For graphical symplectic matroids, this becomes a deletion-contraction recursion that closes up once marked vertices are allowed, and polynomials built from enveloping matroids agree with it exactly for sheltering envelopes.