Independent Set, Chromatic Number, Non-Uniform Coloring of 2-Connected Outerplanar Graphs and Beyond

Grötzsch proved that every triangle-free planar graph has a proper 3-coloring in 1958. One of the generalizations of it is the one that Steinberg and Tovey obtained: Every triangle-free planar graph has a non-uniform [Formula: see text]-coloring [Planar Ramsey numbers, J. Combin. Theory Ser. B 59(2)(1993), 288–296]. Moreover, they proposed the following conjecture: Every planar graph on[Formula: see text] vertices with at most one triangle has a non-uniform [Formula: see text]-coloring. Here, we affirm the conjecture in the situation that every 2-connected outerplanar graph on [Formula: see text] vertices with at most one triangle has a non-uniform [Formula: see text]-coloring by Kempe change.