The total Coloring Conjecture(TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree of its vertices. The conjecture is open for many graphs albeit has been proved for several classes of graphs like complete graphs, trees, cycles, multipartite, graphs of maximum degree $3,4,5$, planar graphs of maximum degree greater than or equal to $6$ and some graphs of high degree. My question pertains to the conjecture for Cayley graphs. In order to prove the result for Cayley graphs, for, say, the Cayley graphs of Abelian groups, Can we take the help of structure theorem for finitely generated abelian groups, like, say, since finitely generated abelian groups are products of cyclic groups upto isomorphism, and, suppose, we are able to prove the TCC for Cayley graphs of cyclic groups with respect to a suitable generator set, then, could we extend it to Cayley graphs of all finitely generated abelian groups with generating set as the cartesian product of the individual generating sets? Any thoughts on this approach? Thanks beforehand.

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