On a construction of graphs with high chromatic capacity and large girth

The chromatic capacity of a graph G , χ C A P ( G ) , is the largest integer k such that there is a k -colouring of the edges of G such that when the vertices of G are coloured with the same set of colours, there are always two adjacent vertices that are coloured with the same colour as that of the edge connecting them. It is easy to see that χ C A P ( G ) ≤ χ ( G ) − 1 . In this note we present a construction based on the idea of classic construction due to B. Descartes for graphs G such that χ C A P ( G ) = χ ( G ) − 1 and G does not contain any cycles of length less than q for any given integer q .