Size of largest complete graph made by contracting edges of a given graph
A graph with four connected subgraphs that, when contracted, form a complete graph. It has no five-vertex complete minor by Wagner's theorem, so its Hadwiger number is exactly four.
In graph theory, the Hadwiger number of an undirected graph G is the size of the largest complete graph that can be obtained by contracting edges of G.
Equivalently, the Hadwiger number h(G) is the largest number n for which the complete graph Kn is a minor of G, a smaller graph obtained from G by edge contractions and vertex and edge deletions. The Hadwiger number is also known as the contraction clique number of G[1] or the homomorphism degree of G.[2] It is named after Hugo Hadwiger, who introduced it in 1943 in conjunction with the Hadwiger conjecture, which states that the Hadwiger number is always at least as large as the chromatic number of G.
The graphs that have Hadwiger number at most four have been characterized by Wagner (1937). The graphs with any finite bound on the Hadwiger number are sparse, and have small chromatic number. Determining the Hadwiger number of a graph is NP-hard but fixed-parameter tractable.
In graph theory, the Hadwigernumber of an undirected graph G is the size of the largest complete graph that can be obtained by contracting edges of G...
on properties that it shares with a different graph parameter, the Hadwigernumber. Later it was again rediscovered by Neil Robertson and Paul Seymour (1984)...
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry...
known as the Hadwiger conjecture or Hadwiger's conjecture. They include: Hadwiger conjecture (graph theory), a relationship between the number of colors...
H-minor-free if it does not have a minor isomorphic to H. Hadwiger 1. Hugo Hadwiger 2. The Hadwigernumber of a graph is the order of the largest complete minor...
have H as a minor may be formed by gluing together simpler pieces, and Hadwiger's conjecture relating the inability to color a graph to the existence of...
snark in modern terminology) must be non-planar. In 1943, Hugo Hadwiger formulated the Hadwiger conjecture, a far-reaching generalization of the four-color...
problems concerning the chromatic number of graphs include the Hadwiger conjecture stating that every graph with chromatic number k has a complete graph on k...
measured in full circles, is the Euler characteristic of the polyhedron. Hadwiger's theorem characterizes the Euler characteristic as the unique (up to scalar...
2151 Hadwiger, provisional designation 1977 VX, is a Marian asteroid from the central region of the asteroid belt, approximately 15 kilometers in diameter...
October 1960 Mathematical Games column. The chromatic number problem, also now known as the Hadwiger–Nelson problem, was a favorite of Paul Erdős, who mentioned...
causes. As an amateur mathematician, he has contributed to the study of the Hadwiger–Nelson problem in geometric graph theory, making the first progress on...
forbidden induced tree The Hadwiger conjecture relating coloring to clique minors The Hadwiger–Nelson problem on the chromatic number of unit distance graphs...
into a bounded number of orthoschemes? (more unsolved problems in mathematics) In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex...
role in several other aspects of graph minor theory: linkless embedding, Hadwiger's conjecture, YΔY-reducible graphs, and relations between treewidth and...
formed from the utility graph K3,3 by subdividing one of its edges. The Hadwiger–Nelson problem asks how many colors are needed to color the points of the...
volume of the ( n − j ) {\displaystyle (n-j)} -dimensional unit ball. Hadwiger's theorem asserts that every valuation on convex bodies in R n {\displaystyle...
3 {\displaystyle n^{4/3}} . The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance...
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