Chapter 5. Theory of Infinite Graphs

Abstract

Publisher SummaryThis chapter provides the definitions of a graph, an independent set, a subgraph, and a circuit. The definitions of coloring number and chromatic number are also discussed. The chromatic number gives the smallest number of colors needed to paint the vertices of graph G so that no two vertices of the same color are joined by an edge. The chapter provides a proof that the coloring number and the chromatic number are related by the inequality Chr(G) ≤ Col(G). The chapter presents a couple of lemmas that present the Erdös–Hajnal example of the graph without odd circuits. The partition symbol α → (β)1γ is used for ordinal numbers α and β, which by definition means that whenever a set, well ordered with order type α, is partitioned into γ classes, at least one of the classes has order type at least β. The chapter discusses a theorem that showsthat any graph with uncountable chromatic number contains even circuits of any length.