The minimum number of edges in a 4-critical graph that is bipartite plus 3 edges

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Abstract

Rödl and Tuza proved that sufficiently large (k+1)-critical graphs cannot be made bipartite by deleting fewer than (k2) edges, and that this is sharp. Chen, Erdős, Gyárfás, and Schelp constructed infinitely many 4-critical graphs obtained from bipartite graphs by adding a matching of size 3 (and called them (B+3)-graphs). They conjectured that every n-vertex (B+3)-graph has much more than 5n/3 edges, presented (B+3)-graphs with 2n3 edges, and suggested that perhaps 2n is the asymptotically best lower bound. We prove that indeed every (B+3)-graph has at least 2n3 edges.

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