Every matroid is a submatroid of a uniformly dense matroid Original Research Article

For a graph G with at least one edge, define d(G) = |E(G)||V(G)| and m(G) = maxH ⊆ G d(H). Karoński and Ruciński (1982) conjectured that every connected graph G is a subgraph of a graph G′ with m(G′) = d(G′) = m(G). This conjecture has been proved by Györi et al. (1985) and, independently by Payan (1986). The following is related. Define g(G)= |E(G)||V(G)| − 1 and λ(G) = maxH ⊆ G d(H). Payan (1986) proves that every connected graph G is a subgraph of a graph G′ with g(G′) = γ(G′) = γ(G). In this paper, we shall show that both theorems above are related by matroid elongations, and we shall also extend these results to their versions in binary matroids and regular matroids.