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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.

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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. $χ(T) ≤ Δ(G) + 2$.

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  • $\begingroup$ yes, that is what I am pointing at. So is there a way to prove the total coloring conjecture using the structure of line graphs? $\endgroup$ – vidyarthi Aug 22 '19 at 7:22
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    $\begingroup$ Well, the problem is open. $\endgroup$ – LeechLattice Aug 22 '19 at 7:23

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