Extremální grafy předepsaného stupně a obvodu

Abstract

In this Ph.D. thesis we investigate the Cage problem. The goal of the Cage problem is to find a smallest graph with regularity k and girth g, so called (k,g)-graphs. We use variety of different methods for finding the (k,g)-graphs.

In Chapter 3 we present results known from the previous research. We follow up on this research in Chapter 4. We first investigate spectra of orders for (k,g)-graphs in Section 4.1. We show full spectra of feasible orders for (2,g)-, (k,3)-, (3,5)- and (3,6)-graphs and partial spectra of orders for (k,4)-graphs. Section 4.1 is based on articles [30, 42]. We then use results from Section 4.1 in Section 4.2 where we try to find new smaller (k,g)-graphs than currently best known graphs using generalized truncation. Also, we use generalized truncation to seek new (k,g)-graphs which then might be possible to shrink down. Section 4.2 is based on article [43]. Sections 4.3 and 4.4 demonstrate how to remove vertices from (k,g)-graphs or how to reconnect vertices in (k,g)-graph while preserving regularity and girth. Though this approach has not brought us any new smaller (k,g)-graphs, we still believe that this approach will bring new results in future and deliver smaller (k,g)-graphs. Sections 4.3 and 4.4 are based on submitted article [22]. In Chapter 5 we explore the properties of \((k,g)\)-graphs which are based on geometric designs. We have been able to determine all the major properties of these (k,g)-graphs. Chapter 5 is based on article [41].

In Chapter 6 we investigate properties of edge-girth-regular graphs. We determine some general properties of these graphs and also improve some lower bounds with the aim of finding conditions that determine whether given edge-girth-regular graph can or cannot exist based on given parameters k,g,lambda. We also determine the smallest possible values of orders for edge-girth-regular graphs with parameters n(4,3,1) and n(4,4,lambda), lambda in {1,2,3,5,6,9}. Chapter 6 is based on article [15].

In Chapter 7 we outline possibilities for future research in the area of (k,g)-graphs. These possibilities are based on some of our ideas and we present a couple of conjectures that may move the research forward in the area of (k,g)-graphs.