A new 4-chromatic edge critical Koester graph

Abstract

Let S be a decomposition of a simple 4-regular plane graph into edge-disjoint cycles such that every two adjacent edges on a face belong to different cycles of S . Such graphs, called Gr¨otzsch–Sachs graphs, may be considered as a result of a superposition of simple closed curves in the plane with tangencies disallowed. Koester studied the coloring of Gr¨otzsch– Sachs graphs when all curves are circles. In 1984, he presented the first example of a 4-chromatic edge critical plane graph of order 40 formed by 7 circles. In the present paper, a new 4-chromatic edge critical graph generated by circles in the plane is presented.