A graph is considered to be totally colored when one color is assigned to each vertex and to each edge so that no adjacent or incident vertices or edges bear the same color. The \textit{total chromatic number} of a graph is the least number of colors required to totally color a graph. This paper focuses on $k$-regular graphs, whose symmetry and regularity allow for a closer look at general total coloring strategies. Such graphs include the previously defined M\"obius ladder, which has a total chromatic number of 5, as well as the newly defined bird's nest, which is shown to have a total chromatic number of 4. Furthermore, a total 4-coloring of the Petersen graph is examined and the total $(k+1)$-colorings of $k$-regular graphs is discussed. More specifically, it is proposed that any $(k+1)$-coloring of any $k$-regular graph is inherently equitable for all $3\leq k\leq5$ given a bound on the order of the graph. That is to say that every color is used no more than one time more than any other color when totally coloring the graph.

Author Bio

Bryson Stemock graduated from Youngstown State University in May of 2019 with a Bachelor of Science degree in Mathematics and Physics & Astronomy. He is currently employed at New Mexico State University as a graduate assistant, where he is pursuing a Ph.D. in Astronomy.