Totally Critical Even Order Graphs

A graph is totally critical if it is Type 2, connected, and the removal of any edge reduces the total chromatic number. A good characterization of all totally critical graphs is unlikely as Sanchez-Arroyo showed that determining the total chromatic number of a graph is an NP-hard problem. In this paper we show that if the Conformability Conjecture is correct, then totally critical graphs of even order with maximum degree at least one half of their order are characterized by a simple equation involving the order, maximum degree, and deficiency of the graph and the edge independence number of the complement. We also show that if the maximum degree Δ of a non-conformable graph G of even order satisfies 12 |V|⩽Δ⩽34 |V|−1 then G is regular and has a simple structure.