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[5] For certain graphs, even fewer than Δ colors may be needed. Δ − 1 colors suffice if and only if the given graph has no Δ-clique, provided Δ is large enough. [6] For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(Δ/log Δ) colors suffice. [7]
If k is sufficiently large, it is known that G has to be 1-factorable: If k = 2n − 1, then G is the complete graph K 2n, and hence 1-factorable (see above). If k = 2n − 2, then G can be constructed by removing a perfect matching from K 2n. Again, G is 1-factorable. Chetwynd & Hilton (1985) show that if k ≥ 12n/7, then G is 1-factorable.
An earlier conjecture of Gabriel Andrew Dirac (1964), proven in 2001 by Wolfgang Mader, states that every n-vertex graph with at least 3n − 5 edges contains a subdivision of K 5. Because planar graphs have at most 3n − 6 edges, the graphs with at least 3n − 5 edges must be nonplanar. However, they need not be 5-connected, and 5-connected ...
If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree with n edges. [7] This is known to be true for sufficiently ...
A path graph or linear graph of order n ≥ 2 is a graph in which the vertices can be listed in an order v 1, v 2, …, v n such that the edges are the {v i, v i+1} where i = 1, 2, …, n − 1. Path graphs can be characterized as connected graphs in which the degree of all but two vertices is 2 and the degree of the two remaining vertices is 1.
An example graph, with 6 vertices, diameter 3, connectivity 1, and algebraic connectivity 0.722 The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1]
However, the maximum list chromatic number of planar graphs is 5, not 4, so the extension fails already for -minor-free graphs. [10] More generally, for every t ≥ 1 {\displaystyle t\geq 1} , there exist graphs whose Hadwiger number is 3 t + 1 {\displaystyle 3t+1} and whose list chromatic number is 4 t + 1 {\displaystyle 4t+1} .
A particle swarm searching for the global minimum of a function. In computational science, particle swarm optimization (PSO) [1] is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality.