New restrictions on defective coloring with applications to steinberg-type graphs

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Steinberg-type graphs, those planar graphs containing no 4-cycles or 5-cycles, became well known with the 1976 Steinberg Conjecture which stated that such graphs are properly 3-colorable. Recently, Steinberg’s Conjecture was demonstrated to be false (Cohen-Addad et al. in J Combin Theory Ser B 122: 452–456, 2016). However, Steinberg-type graphs are (3, 0, 0)-defective colorable (Hill et al. in Discrete Math 313:2312–2317, 2013), i. e. of the three colors, two are used properly and any vertex colored with the first color is allowed to be adjacent to up to three other veritces with the same color. In this paper, we introduce a stronger form of defective graph coloring that places limits on the permitted defects in a coloring. Using the strength of this new type of coloring, we prove the current closest result to Steinberg’s original conjecture and show that the counterexample given in Cohen-Addad et al. (J Combin Theory Ser B 122:452–456,2016) is colorable with this stronger form of defective 3-coloring.

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Journal of Combinatorial Optimization