THE PARTITION DIMENSION OF GRAPHS

Diplomsko delo obravnava particijsko dimenzijo grafov in je sestavljeno iz treh poglavij. V prvem poglavju bomo predstavili osnovne pojme iz teorije grafov in spoznali bomo štiri najbolj poznane produkte grafov, s poudarkom na kartezičnem produktu. Drugo poglavje bomo namenili predstavitvi dveh, za nas najbolj pomembnih dimenzij grafov. To sta metrična in particijska dimenzija grafov. Najprej bomo definirali metrično dimenzijo grafov in spoznali njene lastnosti. Nato se bomo posvetili particijski dimenziji grafov in njenim lastnostim. Pri obeh dimenzijah bomo za boljšo predstavitev podali tudi nekaj primerov. Na koncu tega poglavja pa si bomo še pogledali povezanost omenjenih dveh dimenzij. V zadnjem poglavju bomo definirali particijsko dimenzijo kartezičnega produkta grafov. Pogledali si bomo zgornjo mejo te dimenzije, nato bomo spoznali njeno povezavo z metrično dimenzijo in na koncu navedli še dva aktualna odprta problema. The diploma paper discusses the partition dimension of graphs and consists of three chapters. In the first chapter the basic terms of the graph theory as well as the four most widely known products of graphs, with the emphasis on the Cartesian product, are presented. The second chapter is devoted to the presentation of the two for us most important dimensions of graphs. These are the metric and the partition dimension of graphs. First of all the metric dimension of graphs and it characteristics are presented. Afterwards the emphasis is on the partition dimension of graphs and its features. For a better understanding of both dimensions some examples are added. At the end of this chapter the connections between the two dimensions are presented. In the last chapter the partition dimension of the Cartesian product of graphs is defined. The upper bound of this dimension and the connections with the metric dimension are also shown. In the end two current open problems are stated.