Fu's Equitable Total Coloring Conjecture

Let $G$ be a graph and let $T(G)$ be its total graph, whose vertices represent the vertices and edges of $G$. An equitable $k$-coloring is a proper $k$-coloring whose color classes differ in size by at most one. Fu's Equitable Total Coloring Conjecture. For every graph $G$, $T(G)$ has an equitable $k$-coloring for each

$$k \geq \max\\{\chi(T(G)), \Delta(G)+2\\}.$$

Here $\chi(T(G))$ is the chromatic number of the total graph. The conjecture extends equitable coloring to total graphs and is known for several classes, including complete bipartite graphs, certain complete multipartite graphs, trees, and certain split graphs; it remains open in general.