A result on the total colouring of powers of cycles
The authors prove the following results: 1. Let \(G=C_n^k\), \(n\equiv0\pmod {k+1}\). Then \(\chi_T(G)\leq\Delta(G)+2\). 2. The graph \(G=C_n^k\), \(n\equiv r\pmod{k+1}\), with \(n\) even and \(r\neq0\), has a \((\Delta(G)+2)\)-total colouring. The authors conjecture: Let \(G=C_n^k\), with \(2\leq k<\lfloor \frac n2\rfloor\). Then \[ \chi_T(G)=\begin{cases} \Delta(G)+2&\text{if }k>\frac n3 -1\text{ and }n\text{ odd,}\\ \Delta(G)+1&\text{otherwise.}\end{cases} \]
scientific article; zbMATH DE number 5142431