3–edge colorable graphs

In this BSc thesis we deal with chromatic index of cubic graphs, where we mainly focus on a significant part of the family of graphs, named generalized Petersen graphs. A graph Γ is said to be k-edge-colorable, if we can color its edges with k colors, so that incident edges are colored with different colors. The smallest such number k is called the chromatic index and it is denoted by χ'(Γ). Due to the fact that generalized Petersen graphs are cubic graphs, Vizing's theorem implies that their chromatic index is either 3 or 4. The results of this BSc thesis represent an important part of the proof, that the famous Petersen graph is the only generalized Petersen graph, which is not 3-edge colorable. In other words, the Petersen graph GP(5,2) is the only generalized Petersen graph, whose chromatic index equals 4

The independence number of a graph

In the master's thesis we are dealing with the independence number of a graph. We show, that the well-known problem 3-SAT is reducible to the corresponding decision problem, the so-called independent set problem, which proves that the independent set problem is NP-complete. We then determine the independence number for different graphs, including some very well known infinite families of graphs like complete graphs, multi-partite complete graphs, cycle graphs, hypercube graphs, etc. In the last part of the thesis we focus on the family of generalized Petersen graphs GP(n,k). Based on their construction it is clear, that n is the upper bound for the independence number for GP(n,k). Moreover, if n is odd, the upper bound is n-1. In the master's thesis we determine the exact value of the independence number for different values of parameter k

The independence number of a graph

V magistrskem delu se ukvarjamo s problemom določanja neodvisnostnega števila grafa. S pomočjo prevedbe problema 3-SAT na pripadajoči odločitveni problem o obstoju neodvisnostne množice dane velikosti najprej pokažemo, da ga uvrščamo med tako imenovane NP-polne probleme. Nato se osredotočimo na določanje neodvisnostnega števila za različne grafe. Določimo ga za nekatere dobro znane družine grafov, kot so polni grafi, polni večdelni grafi, cikli, hiperkocke itd. Posvetimo se tudi znani družini posplošenih Petersenovih grafov GP(n,k). Glede na konstrukcijo te družine je jasno, da je zgornja meja neodvisnostnega števila za GP(n,k) največ n, če pa je n liho število, pa celo največ n-1. V magistrskem delu raziskujemo, kakšna je prava vrednost neodvisnostnega števila za različne vrednosti parametra k in s tem ugotavljamo, kako dobra (oziroma slaba) je omenjena zgornja meja.In the master\u27s thesis we are dealing with the independence number of a graph. We show, that the well-known problem 3-SAT is reducible to the corresponding decision problem, the so-called independent set problem, which proves that the independent set problem is NP-complete. We then determine the independence number for different graphs, including some very well known infinite families of graphs like complete graphs, multi-partite complete graphs, cycle graphs, hypercube graphs, etc. In the last part of the thesis we focus on the family of generalized Petersen graphs GP(n,k). Based on their construction it is clear, that n is the upper bound for the independence number for GP(n,k). Moreover, if n is odd, the upper bound is n-1. In the master\u27s thesis we determine the exact value of the independence number for different values of parameter k