Theorem: If $G$ is a graph in which all its cycle lengths are divisible by 4, then $sq(G) \le \Delta^2(G)$.
$sq(G)$ denote strong chromatics index and $\Delta(G)$ denote maximum degree of graph $G$
proof of this theorem is on page 209, theorem 6 https://pdfs.semanticscholar.org/c62d/02fea7953aeecf64e27db215281f262cd0f7.pdf
my question is what does it mean that $G$ has no cycle with a diagonal? and why?