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Infinite graphs with such an orientation do not always have a low-degree vertex (for instance, Bethe lattices have = but arbitrarily large minimum degree), so this argument requires the graph to be finite. But the De Bruijn–Erdős theorem shows that a (+)-coloring exists even for infinite graphs. [5]
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.
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 the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph. [1]
In mathematics and computer science, graph theory is the study of graphs, ... for infinite graphs because many of the arguments fail in the infinite case.
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 ...
Pages in category "Infinite graphs" The following 15 pages are in this category, out of 15 total. ... End (graph theory) H. Hadwiger–Nelson problem; Halin's grid ...
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.
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