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Halin (1965) defined a thick end of a graph to be an end that contains infinitely many rays that are pairwise disjoint from each other. The hexagonal tiling of the plane. An example of a graph with a thick end is provided by the hexagonal tiling of the Euclidean plane. Its vertices and edges form an infinite cubic planar graph, which contains ...
The infinite graphs that contain Eulerian lines were characterized by Erdõs, Grünwald & Weiszfeld (1936). For an infinite graph or multigraph G to have an Eulerian line, it is necessary and sufficient that all of the following conditions be met: [18] [19] G is connected. G has countable sets of vertices and edges. G has no vertices of (finite ...
The Rado graph, as numbered by Ackermann (1937) and Rado (1964).. In the mathematical field of graph theory, the Rado graph, Erdős–Rényi graph, or random graph is a countably infinite graph that can be constructed (with probability one) by choosing independently at random for each pair of its vertices whether to connect the vertices by an edge.
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.
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. [44]
Aho et al. provide the following example to show that in infinite graphs, even when the graph is acyclic, a transitive reduction may not exist. Form a graph with a vertex for each real number, with an edge whenever < as real numbers. Then this graph is infinite, acyclic, and transitively closed.
Kőnig's 1927 publication. Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. [1] It gives a sufficient condition for an infinite graph to have an infinitely long path.
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. [1] If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. [2] For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3.