Flow reconfiguration with one modulus for all graphs, and Cereceda's conjecture for bipartite 2-degenerate graphs

This paper answers a question of Esperet, Hendrey, Lagoutte, Marseloo, Norin and Steiner on flow reconfiguration and makes inroads into Cereceda's recolouring conjecture for d = 2. Parts I and II are about flows. For a finite multigraph G and an integer k ≥ 2, let Mk(G) be the graph on the nowhere-zero ℤk-flows of G, and Ik(G) the graph on the integer flows F of G with 1 ≤ |F(e)| ≤ k − 1 on every edge; two flows are adjacent if the support of their difference is a cycle. We prove that Mk(G) and Ik(G) are connected for every finite bridgeless loopless multigraph G if k = 40, if k = 48, or if k is a multiple of 4 with k ≥ 52. This answers their Problem 1.2. The same holds for k = 66, for every prime power pN with N ≥ 6, and for a set of moduli of natural density one; these results do not cover every k ≥ 40. A master theorem for finite abelian groups gives the connectivity of the flow graph F(G, Γ) for every abelian group Γ of order 40, and a binary form of it for every abelian group of exponent dividing 4 and order at least 32, in particular 𝔽25 (Esperet et al. proved 𝔽28). Part III is about colourings: two proper k-colourings of a graph G are adjacent if they differ at exactly one vertex. Cereceda conjectured that this graph has diameter O(n2) for every d-degenerate graph on n vertices and k = d + 2; the case d = 2 is open. For four colours we prove the diameter bound 2n(n + 1) for every bipartite graph of degeneracy at most 2, with the order n2 sharp in this class, also at girth 6; the bound 5n(n + 1) when an independent set of vertices of degree at most 2 is added; and, with a computer-verified certificate, 4n(n + 1) for every graph of cone width at most 2, a class that contains all graphs of treewidth at most 2 and some graphs of treewidth 3 with triangles.