Hadwiger number

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 K_n 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 or the homomorphism degree of G. 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 . 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. A graph G has Hadwiger number at most two if and only if it is a forest, for a three-vertex complete minor can only be formed by contracting a cycle in G. A graph has Hadwiger number at most three if and only if its treewidth is at most two, which is true if and only if each of its biconnected components is a series–parallel graph. Wagner's theorem, which characterizes the planar graphs by their forbidden minors, implies that the planar graphs have Hadwiger number at most four. In the same paper that proved this theorem, also characterized the graphs with Hadwiger number at most four more precisely: they are graphs that can be formed by clique-sum operations that combine planar graphs with the eight-vertex Wagner graph. The graphs with Hadwiger number at most five include the apex graphs and the linklessly embeddable graphs, both of which have the complete graph K_6 among their forbidden minors. Every graph with n vertices and Hadwiger number k has O(nk \sqrt{ \log k}) edges. This bound is tight: for every k, there exist graphs with Hadwiger number k that have \Omega (nk \sqrt{ \log k}) edges.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (8)

Related people (2)

Related concepts (2)

Distributed Coloring of Graphs with an Optimal Number of Colors

Etienne Michel François Bamas

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e. with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on ...

SCHLOSS DAGSTUHL, LEIBNIZ CENTER INFORMATICS2019

Coloring Intersection Graphs of Arc-Connected Sets in the Plane

Bartosz Maria Walczak, Michal Lason

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line L if the intersection of any member with L is a nonempty segment. It is proved that the intersection graphs of simple families of com ...

Springer2014

Triangle-free intersection graphs of line segments with large chromatic number

Bartosz Maria Walczak, Michal Lason

In the 1970s Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer k we construct a triangle-free fam ...

Academic Press Inc Elsevier Science2014

Clique-sum

In graph theory, a branch of mathematics, a clique-sum is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology. If two graphs G and H each contain cliques of equal size, the clique-sum of G and H is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then possibly deleting some of the clique edges. A k-clique-sum is a clique-sum in which both cliques have at most k vertices.

Apex graph

In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K_5 or K_3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.