Some Problems in Total Graph Theory

Thirty years ago the total chromatic number and the total graph of a graph was introduced and a conjecture was stated in the author’s Ph.D. dissertation. This conjecture is known as the Total Chromatic Coiyecture (TCC). At this time numerous results concerning these and other total concepts such as total groups, total crossing numbers, and total Ramsey numbers exist in the literature. More specifically, during these decades results concerning different properties of total graphs, such as planarity, reconstructibility, and traversability have been obtained, and some parameters, including (vertex) connectivity, edge connectivity, arboricity, and eigenvalues, of such graphs have been studied. Some total concepts have been generalized, and some have been applied to other areas. In recent years a lot of attention has been paid to TCC and tremendous effort is being used toward its settlement. In this brief expository article we confine ourselves to some concepts and questions which are tangible even by some bright undergraduate students. As far as references are concerned, as usual, there are a lot of duplications. For the sake of briefness we have to be selective, and, with apology, we eliminate a lot of articles and results.