# The Total Graph is similar to a line graph

Consider the total graph of a regular graph. From the structure, it seems that it has a similar structure to the line graph ( two different sub-cliques joining at a single point) except that, in addition, there is an edge between two different sub-cliques. Note that the sub-cliques in the total graph are the just the sub-cliques which are present in the line graph of the graph with a vertex added to each sub-clique make it adjacent to all the vertices of a particular sub-clique, thereby increasing the order of the sub-cliques by one. This means that the clique number of the total graph of a graph is one more than the clique number of the line graph of the same graph.

This leads me to infer that the total graph of a graph has similar chromatic properties as a line graph of a graph. Specifically, I expect the total graph of a graph to satisfy a similar bound to Vizing's theorem for line graphs. Will this be true? Thanks beforehand.

It depends on what you mean by "similar". It's known that $$χ(T) ≤ Δ(G) + 10^{26}$$, where $$T$$ is the total graph of $$G$$. As the graph $$C_4$$ have $$Δ=2$$ and $$χ(T)=4$$, it follows that the constant term is at least $$2$$. The Total coloring conjecture states that this is exactly the case, i.e. $$χ(T) ≤ Δ(G) + 2$$.