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Cycles of all possible lengths in the graph of an octahedron, showing it to be pancyclic.. In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph. [1]
Triangle-free graphs: Triangle K 3: Induced subgraph Definition Planar graphs: K 5 and K 3,3: Homeomorphic subgraph Kuratowski's theorem: K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph minor Auer et al. (2013) [2] Graphs of ...
Proof without words that a hypercube graph is non-planar using Kuratowski's or Wagner's theorems and finding either K 5 (top) or K 3,3 (bottom) subgraphs. If is a graph that contains a subgraph that is a subdivision of or ,, then is known as a Kuratowski subgraph of . [1]
Line graphs may be characterized in terms of nine forbidden subgraphs; [2] the claw is the simplest of these nine graphs. This characterization provided the initial motivation for studying claw-free graphs. [1] The complement of any triangle-free graph is claw-free. [3] These graphs include as a special case any complete graph.
A graph is chordal if and only if it has a perfect elimination ordering. [3] Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth-first search. This algorithm maintains a partition of the vertices of the ...
4-connected planar graphs are always Hamiltonian by a result due to Tutte, and the computational task of finding a Hamiltonian cycle in these graphs can be carried out in linear time [18] by computing a so-called Tutte path. Tutte proved this result by showing that every 2-connected planar graph contains a Tutte path.
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The possibility of a graph with these parameters was already suggested in 1969 by Norman L. Biggs, [6] and its existence noted as an open problem by others before Conway. [3] [7] [8] [9] Conway himself had worked on the problem as early as 1975, [7] but offered the prize in 2014 as part of a set of problems posed in the DIMACS Conference on Challenges of Identifying Integer Sequences.