An Investigation of Colour-Critical Graphs with Complements of Low Connectivity

Abstract

Publisher Summary This chapter investigates color-critical graphs with complements of low connectivity. A k -chromatic graph Γ is called v -critical if all vertices x of the graph are critical. It is called e -critical if it is connected and all edges e of the graph are critical. It follows from these definitions that an e -critical graph is also v -critical. The critical 3-chromatic graphs are the odd circuits. It seems hopeless to try to determine the structure of all critical graphs with chromatic number 4 or more. A graph is the complement of a v -critical ( e -critical) graph Γ if and only if each connected component of is the complement of a v -critical ( e -critical) graph. This result gives a method for constructing new critical graphs from known ones.