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In 1960, Dirac posed the conjecture that r-connected 4-critical graphs exist for every r ! 3. In 1989, Erdo´s conjectured that for every r ! 3 there exist r-regular 4-critical graphs. In this paper, a technique of constructing r-regular r-connected vertex-transitive 4-critical graphs for even r ! 4 is presented. Such graphs are found for r ¼ 6, 8, 10. ß

It is proved that by deleting at most 5 edges every planar (simple) graph of order at least 2 can be reduced to a graph having a non-trivial automorphism. Moreover, the bound of 5 edges cannot be lowered to 4.

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