Edge $4$-critical Koester graph of order $28$

A Koester graph $G$ is a simple $4$-regular plane graph formed by the superposition of a set $S$ of circles in the plane, no two of which are tangent and no three circles have a common point. Crossing points and arcs of $S$ correspond to vertices and edges of $G$, respectively. A graph $G$ is edge critical if the removal of any edge decreases its chromatic number. A $4$–chromatic edge critical Koester graph of order $28$ generated by intersection of six circles is presented. This improves an upper bound for the smallest order of such graphs. The previous upper bound was established by Gerhard Koester in 1984 by constructing a graph with $40$ vertices.