The strong chromatic index of graphs

The strong chromatic index of graphs Mohammad Mahdian Master of Science Graduate Department of Computer Science University of Toronto A strong edge colouring of a graph G is an assignment 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 Erd os and Ne set ril conjectured that the strong chromatic index of every graph of maximum degree is at most In this thesis we present a survey of known results related to strong edge colourings and an introduction to the probabilistic method We will use the probabilistic method to prove that the strong chromatic index of a C free graph i e a graph which does not contain a cycle as a subgraph of maximum degree is at most o ln This implies that the conjecture of Erd os and Ne set ril is true for C free graphs with large maximum degree We will show that our bound is asymptotically the best possible up to a constant multiple Also we will investigate the algorithmic aspects of the strong edge colouring problem and will prove that it is NP complete even in a very restricted setting Finally we present a list of open problems and conjectures related to strong edge colourings

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