A Sharp Three-Fifths Bound for Seymour Vertices in Three-Regular Oriented Graphs

Let D be a finite simple oriented graph with indegree and outdegree three at every vertex. A Seymour vertex has at least as many exact second out-neighbors as first out-neighbors. We prove that at least ceil(3|V(D)|/5) vertices are Seymour vertices. If B is the set of the remaining vertices, the sum of |N^{++}(v)|-3 over all vertices is at least ceil(|B|/2). Both constants are sharp, even for strongly connected graphs.

We classify all graphs attaining the exact, unrounded three-fifths proportion. They are expansions of loopless two-in/two-out directed multigraphs: replace each vertex with a directed triangle and each arc with a new vertex joined completely to its two endpoint triangles. An exact surplus formula then characterizes simultaneous equality in both bounds. The proofs use local arc capacities, indegree saturation and equality in an incidence count. The package includes the six-page English manuscript, standalone LaTeX and reproducible finite checks.

This is a complete theorem for the three-regular class associated with AIM-COMBINATORICS-0183 in UnsolvedMath v1.6.0. It does not resolve the mean second-neighborhood inequality for arbitrary Eulerian oriented graphs, including mixed degrees at most three. The source record remains partial. The manuscript is AI-assisted, self-audited and unrefereed; independent review, formal verification and absolute priority are not claimed.