Exact 6-cut rigidity and small-order superconnectivity for the 6-regular case of Dirac's k=4 problem

Abstract

Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphism) on n=13, whose 13 critical edges form a Hamilton cycle; hence any 6-regular (4,1)-graph has at least 16 vertices. Theorem B: in a 6-regular (4,1)-graph every 6-edge-cut is either the edge star of a vertex or has both shores of size at least 15; consequently every 6-regular (4,1)-graph on at most 29 vertices is super-6-edge-connected. Theorem C (all sizes): no shore of a nontrivial 6-edge-cut in a 6-regular (4,1)-graph induces a bipartite graph; more generally, a shore whose deficiency is concentrated on two vertices forces them to receive equal colours in every proper 3-colouring. The proof of Theorem B rests on an exact classification of the 3x3 cut matrices of 6-edge-cuts in (4,1)-graphs (exactly 21 matrices, five types up to row/column permutations) together with a boundary-shortfall lemma; the unique near-miss is K_{3,3,3} minus a rainbow 3-matching. Several supporting lemmas are machine-checked in Lean 4/Mathlib.