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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 Hajós join of two copies of P∗ .