Edge-critical graphs from the Möbius strip
Let χ'(G) denote the chromatic index of G and denote by G-e the graph obtained by removing an edge e from G. The graph G is said to be Δ-critical if χ'(G)=Δ+1 and χ'(G-e)=Δ for any edge e in E(G). We are interested in critical graphs of even order.
It has been proved that there are no critical graphs of even order n≤ 10 and that there are no 3-critical graphs of order 12 and 14 (see ). Based on these results, Jakobsen formulated the famous critical graph conjecture, which claims that there are no critical graphs of even order (see ). A similar conjecture was made by Beineke and Wilson .
The graphs found by Goldberg , Fiol , Chetwynd and Wilson  disproved the conjecture. Goldberg constructed a 3-critical simple graph of order 22; Fiol found two 4-critical simple graphs of order 18 and 30; Chetwynd and Wilson found a 4-critical graph of order 16 with multiple edges.
Following the approach presented in , we construct infinite families of critical graphs of even order. Our method is based on a suitable identification of vertices which resembles the topological identification yielding the Möbius strip from a rectangular strip. By this method, we can also obtain the counterexamples found by Goldberg, Fiol, Chetwynd and Wilson.
 L.W. Beineke, R.J. Wilson, On the edge-chromatic number of a graph, Discrete Math. 5 (1973), 15-20.
 A.G. Chetwynd, R.J. Wilson, The rise and fall of the critical graph conjecture, J. Graph Theory 7 (1983), 153-157.
 M.A. Fiol, 3-grafos criticos, Doctoral dissertation, Barcelona University, Spain (1980).
 S. Fiorini, R.J. Wilson, Edge-colourings of graphs, Research Notes in Mathematics, Vol. 16, Pitman, London (1977).
 M.K. Goldberg, On graphs of (maximum) degree $3$ with chromatic index 4 (Russian), Bull. Acad. Sci. Georgia, SSR, 93 (1979), 29-31.
 I.T. Jakobsen, On critical graphs with chromatic index 4, Discrete Math. 9 (1974), 265-276.
 A. Vietri, An analogy between edge colourings and differentiable manifolds, with a new perspective on 3-critical graphs, Graphs Comb. 31 (2015), 2425-2435.