6 problems

Let $G$ be a graph, and let $T(G)$ be its total graph, whose vertices are the vertices and edges of $G$, with adjacency given by adjacency, incidence, or edge adjacency in $G$. The…

Let $G$ be a finite multigraph, and let $T(G)$ be its total graph. Write $\chi(T(G))$ for the chromatic number and $\chi_l(T(G))$ for the list chromatic number. Borodin–Kostochka–W…

Let $k\geq2$ be a fixed positive integer and let $G$ be a graph. $(k)$-total coloring conjecture. … This is presented as a weaker version of the total coloring conjecture obtained…

Let $T$ be a total graph, and let $\chi_\ell(T)$ denote the smallest $m$ such that $T$ is $m$-choosable. Let $\Delta(T)$ be the maximum degree of $T$, and use equitable $k$-choosab…

Let $G$ be a finite simple graph, let $T(G)$ be its total graph, and let $\chi_l(T(G))$ denote the list chromatic number of $T(G)$. A graph is equitably $k$-choosable if every list…

Total-graph coloring conjecture.