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In the mathematics of infinite graphs, an end of a graph represents, intuitively, a direction in which the graph extends to infinity. Ends may be formalized mathematically as equivalence classes of infinite paths, as havens describing strategies for pursuit–evasion games on the graph, or (in the case of locally finite graphs) as topological ends of topological spaces associated with the graph.
In graph theory, a branch of mathematics, Halin's grid theorem states that the infinite graphs with thick ends are exactly the graphs containing subdivisions of the hexagonal tiling of the plane. [1] It was published by Rudolf Halin ( 1965 ), and is a precursor to the work of Robertson and Seymour linking treewidth to large grid minors , which ...
In infinite graph theory, an end is defined slightly differently, as an equivalence class of semi-infinite paths in the graph, or as a haven, a function mapping finite sets of vertices to connected components of their complements.
A stronger but unbalanced infinite form of Ramsey's theorem for graphs, the Erdős–Dushnik–Miller theorem, states that every infinite graph contains either a countably infinite independent set, or an infinite clique of the same cardinality as the original graph. [45]
In graph theory, the De Bruijn–Erdős theorem relates graph coloring of an infinite graph to the same problem on its finite subgraphs. It states that, when all finite subgraphs can be colored with c {\displaystyle c} colors, the same is true for the whole graph.
Menger's theorem holds for infinite graphs, and in that context it applies to the minimum cut between any two elements that are either vertices or ends of the graph . The following result of Ron Aharoni and Eli Berger was originally a conjecture proposed by Paul Erdős , and before being proved was known as the Erdős–Menger conjecture .
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A ray, in an infinite graph, is an infinite simple path with exactly one endpoint. The ends of a graph are equivalence classes of rays. reachability The ability to get from one vertex to another within a graph. reachable Has an affirmative reachability. A vertex y is said to be reachable from a vertex x if there exists a path from x to y ...