The Strong Chromatic Index of Graphs

The strong chromatic index of graphs Mohammad Mahdian Master of Science Graduate Department of C o m p t e r Science University of Toronto 2000 A strong edge-colouring of a graph G is an assignrnent of colours to the edges of G such that every colour class is an induced matching. The minimum number of colours in such a colouring is called the strong chromatic index of G. In 1985, Erdos and Nesetfil conjectured that the strong chromatic index of every graph of maximum degree A is at most $A2. In this thesis, we present a survey of knotvn results related to strong edgecolourings, and an introduction to the probabilistic method. Vie will use the probabilistic rnethod to prove that the strong chromatic index of a CJree graph (Le., a graph which does not contain a 4-cycle as a subgraph) of maximum degree A is a t most (2 + o(l)) &. This implies that the conjecture of Erdos and NeSetfil is true for Ca-free graphs with large maximum degree. We will show that our bound is asymptotically the best possible, up to a constant multiple. -41~0, we wili investigate the algoritlimic aspects of the strong edge-colouring problem, and ivill prove that it is NP-complete even in a very restricted setting. Finally, ive present a list of open problems and conjectures related to strong edge-colourings.