Connectivity of Minimum Non-5-injectively Colorable Planar Cubic Graphs

Abstract

Suppose that G is a planar cubic graph with $$\chi _i(G)>5$$. We show that if $$\chi _i(H)<\chi _i(G)$$ for each planar cubic graph H of order less than G, then G is either a 3-connected simple planar cubic graph, or a planar graph obtained from a simple cubic 3-connected planar graph by adding some earrings. This shows that a minimum non-5-injectively colorable simple planar cubic graph must be 3-connected.