10 problems

A graph $G$ has vertex set $V(G)$ and edge set $E(G)$. A total-colouring assigns colours to the vertices and edges of $G$ so that no adjacent or incident elements receive the same…

Khennoufa–Togni conjecture. Only finitely many graphs $C_n(1,3)$ are Type II; all remaining graphs $C_n(1,3)$ are Type I.

Total colouring conjecture. For every simple graph $G$,

Let $G$ be a graph with maximum degree $\Delta$. In the natural list version of neighbour sum distinguishing total colouring, colours are chosen from arbitrary lists of real number…

Let $G$ be a graph with maximum degree $\Delta$, and let $\chi”_{\sum}(G)$ denote the least number of colours in a proper total colouring whose weighted degrees distinguish adjacen…

Let $G$ be a graph, and let $\chi”(G)$ be its total chromatic number: the least number of colours in a proper total colouring of $G$, where adjacent vertices, incident edges, and e…

For a graph $G$, let $\chi”(G)$ denote its total chromatic number and let $\Delta(G)$ denote its maximum degree. Total colouring conjecture. For all graphs $G$, … This strengthens…

Zhang–Chen–Li–Yao–Lu–Wang's conjecture. For every graph $G$,

Skowronek-Kaziόw's conjecture. For every graph $G$,

Total 1-2 Conjecture. For every graph $G$,