Subcubic Edge-Chromatic Critical Graphs Have Many Edges

We consider graphs G with Δ=3 such that χ′(G)=4 and χ′(G−e)=3 for every edge e, so-called critical graphs. Jakobsen noted that the Petersen graph with a vertex deleted, P∗, is such a graph and has average degree only 83. He showed that every critical graph has average degree at least 83, and asked if P∗ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if G is a subcubic critical graph other than P∗, then G has average degree at least 4617≈2.706. This bound is best possible, as shown by the Hajos join of two copies of P∗.