Minimal Line Distinguishing Colourings in Graphs

Abstract

In [4] it is shown that for a graph G of diameter 2 on an even number n of vertices to be minimally line-distinguishing (MLD-) colourable we must have Λ = n/2, where Λ is the line distingushing chromatic number of G. An easy example is the Peterson graph. For the case in which n is odd the condition for a MLD-colouring that every pair of colours occurs exactly once cannot be met and the best that can be achieved is that Λ = n/2 + 1. In [4] fig.2 an infinite class of diameter 2 graphs on an odd number of vertices for which Λ = n/2 + 1 is constructed. In #2 some results for graphs of diameter 2 are found. In particular a class of diameter 2 graphs having Λ = n/2 is exhibited and it is shown that the Peterson graph is the only strongly regular MLD-colourable graph. In #3 it is shown by construction that a class of regular MLD-colourable graphs of any odd valency p>1 exists.