The planar and bounded-treewidth cases of the GNRS L1-embedding conjecture

Planar paper, Theorem 1.1: a universal constant $C\ge1$ such that every finite connected planar graph with positive edge lengths embeds into $L_1$ with distortion at most $C$; hence the undirected multicommodity flow-cut gap is uniformly bounded on planar graphs. Treewidth companion, Theorem 1.1: for each $k$, a constant $C_k$ (no explicit estimate) for graphs with tree decompositions of bag size at most $k$, hence also for graphs excluding a fixed planar minor. Together they give constant distortion for fixed-parameter almost-embeddable families. Not shown: the full GNRS conjecture for every minor-closed family (closure under clique sums is open), and no explicit value of $C$.

Planar paper, Theorem 1.1: a universal constant $C\ge1$ such that every finite connected planar graph with positive edge lengths embeds into $L_1$ with distortion at most $C$; hence the undirected multicommodity flow-cut gap is uniformly bounded on planar graphs. Treewidth companion, Theorem 1.1: for each $k$, a constant $C_k$ (no explicit estimate) for graphs with tree decompositions of bag size at most $k$, hence also for graphs excluding a fixed planar minor. Together they give constant distortion for fixed-parameter almost-embeddable families. Not shown: the full GNRS conjecture for every minor-closed family (closure under clique sums is open), and no explicit value of $C$.