Some theorems concerning the star chromatic number of a graph (Q1362095)

Let \(d\) be a positive integer satisfying \(m\geq 2d\). An \((m,d)\)-colouring of a graph \(G\) is a function \(f:V\to Z_m\) such that \(|f(u)- f(v)|_m\geq d\) for each edge \(uv\). The star-chromatic number of \(G\) is defined as \(\inf\{m/d: G\) has an \((m,d)\)-colouring\}. The author investgates the star-chromatic number of a graph via the chromatic number, girth, independence number, length of a longest cycle of the graph, respectively.