The History of the Total Chromatic Number Conjecture

Abstract

The total chromatic number conjecture, which has appeared in a few hundred articles and in numerous books thus far, is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V.G. Vizing. Eventually, after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in *The Mathematical Coloring Book* (2009); and stated that, “In my opinion this unquestionably merits the joint credit to Vizing and Behzad.” After checking all the arguments presented and the blames cited, I decided to investigate the controversy stated in this book on my own. My findings, which are presented in this report, specifically signify the following two points:

1. M. Behzad is the sole author of the Total Chromatic Number Conjecture.
2. The wrong referrals provided by numerous authors over the last forty-four years, to indicate Vizing’s authorship, must be brought to the attention of the authors and researchers, by appropriate means, as soon as possible.