Fractional and circular 1-defective colorings of outerplanar graphs.
Zuzana Farkasová, Roman Soták · Australas. J Comb. · 2015
A graph is called fractional ( r s , d ) -defective colorable if its vertices can be colored with r colors in such a way that each vertex receives s distinct colors and has at most d defects (a defect corresponds to the situation when two adjacent vertices are assigned with non-disjoint sets). We show that each outerplanar graph having no triangle faces sharing a vertex is fractional ( 7 3 , 1 ) -defective colorable; moreover, this bound is tight also in the case when the graph has no touching triangles. These results correct the claim in [W. Klostermeyer, Defective circular coloring, Australas. J. Combin. 26 (2002), 21–32] on circular ( 5 2 , 1 ) -defective colorability of outerplanar graphs having no adjacent triangles. Further, we show that if one allows overlapping triangles then one cannot improve on the upper bound of 3 given by the 3-colorability of outerplanar graphs.