Hadwiger conjecture (graph theory)

In graph theory, the Hadwiger conjecture (or Hadwiger's conjecture) states that, if an undirected graph "G" requires "k" or more colors in any vertex coloring, then one can find "k" disjoint connected subgraphs of "G" such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single supervertex produces a complete graph "K_k" on "k" vertices as a minor of "G".

This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. harvtxt|Bollobás|Catlin|Erdős|1980 call it “one of the deepest unsolved problems in graph theory.”

Equivalent forms

An equivalent form of the Hadwiger conjecture (the contrapositive of the form stated above) is that, if there is no sequence of edge contractions (each merging the two endpoints of some edge into a single supervertex) that brings graph "G" to the complete graph "K_k", then "G" must have a vertex coloring with "k" − 1 colors.

Note that, in a minimal "k"-coloring of any graph "G", contracting each color class of the coloring to a single vertex will produce a complete graph "K_k". However, this contraction process does not produce a minor of "G" because the contracted sets of vertices are not in general connected. Hadwiger's conjecture states that there exists a different way of contracting sets of vertices to single vertices, producing a complete graph "K_k", in such a way that all the contracted sets are connected.

If "F_k" denotes the family of graphs having the property that all minors of graphs in "F_k" can be ("k" − 1)-colored, then it follows from the Robertson–Seymour theorem that "F_k" can be characterized by a finite set of forbidden minors. Hadwiger's conjecture is that this set consists of a single forbidden minor, "K_k".

pecial cases

The case where "k" = 2 is trivial: a graph requires more than one color if and only if it has an edge, and that edge is itself a "K"₂ minor. The case "k" = 3 is also easy: the graphs requiring three colors are the non-bipartite graphs, and every non-bipartite graph has an odd cycle, which can be contracted to a 3-cycle, that is, a "K"₃ minor.

In the same paper in which he introduced the conjecture, Hadwiger proved its truth for "k" ≤ 4. The graphs with no "K"₄ minor are the series-parallel graphs and their subgraphs. Each graph of this type has a vertex with at most two incident edges; one can 3-color any such graph by removing one such vertex, coloring the remaining graph recursively, and then adding back and coloring the removed vertex. Because the removed vertex has at most two edges, one of the three colors will always be available to color it when the vertex is added back.

The truth of the conjecture for "k" = 5 implies the four color theorem: for, if the conjecture is true, every graph requiring five or more colors would have a "K"₅ minor and would (by Wagner's theorem) be nonplanar.
Klaus Wagner proved in 1937 that the case "k" = 5 is actually equivalent to the four color theorem and therefore we now know it to be true. As Wagner showed, every graph that has no "K"₅ minor can be decomposed via clique-sums into pieces that are either planar or an 8-vertex Möbius ladder, and each of these pieces can be 4-colored independently of each other, so the 4-colorability of a "K"₅-free graph follows from the 4-colorability of each of the planar pieces.

harvtxt|Robertson|Seymour|Thomas|1993 proved the conjecture for "k" = 6, also using the four color theorem. Thus the conjecture is true for "k" ≤ 6 but it remains unsolved for all "k" > 6.

For "k" = 7, some partial results are known: every 7-chromatic graph must contain either a "K"₇ minor or both a "K"_4,4 minor and a "K"_3,5 minor. [The existence of either a "K"₇ or "K"_3,5 minor was shown by Kawarabayashi, and harvtxt|Kawarabayashi|Toft|2005 proved the existence of either a "K"₇ or "K"_4,4 minor.]

Hadwiger number

The Hadwiger number "h"("G") of a graph "G" is the size "k" of the largest complete graph "K_k" that is a minor of "G" (or equivalently can be obtained by contracting edges of "G"). It is also known as the contraction clique number of "G".harvtxt|Bollobás|Catlin|Erdős|1980.] The Hadwiger conjecture can be stated in the simple algebraic form "χ"("G") ≤ "h"("G") where "χ"("G") denotes the chromatic number of "G". A related concept, the achromatic number of "G", is the size of the largest clique that can be formed by contracting a family of independent sets in "G").

Determining the Hadwiger number of a given graph is fixed-parameter tractible: there is an algorithm for finding the largest clique minor in an amount of time that depends only polynomially on the size of the graph, but exponentially in "h"("G"). [harvtxt|Alon|Lingas|Wahlen|2006.] harvtxt|Alon|Lingas|Wahlen|2006 consider approximation algorithms for finding large clique minors in graphs, and observe that (unless P=NP) polynomial time algorithms can approximate the Hadwiger number significantly more accurately than the best possible approximation to the size of the largest clique in a graph.

It is known that every graph "G" with "h"("G") = "k" has a vertex with at most O(k √log "k") incident edges. [harvtxt|Kostochka|1984. The letter O in this expression invokes big O notation.] By removing this low-degree vertex, coloring the remaining graph, and then adding back the removed vertex and coloring it, one can show that "χ"("G") = O(k √log "k").

Generalizations

György Hajós conjectured that Hadwiger's conjecture could be strengthened to subdivisions rather than minors: that is, that every graph with chromatic number "k" contains a subdivision of a complete graph "K_k". Hajós' conjecture is true for "k" ≤ 4, but harvtxt|Catlin|1979 found counterexamples to this strengthened conjecture for "k" ≥ 7; the cases "k" = 5 and "k" = 6 remain open. [harvtxt|Yu|Zickfeld|2006.] harvtxt|Erdős|Fajtlowicz|1981 observed that Hajós' conjecture fails badly for random graphs: for any ε > 0, in the limit as the number of vertices, "n", goes to infinity, the probability approaches one that a random "n"-vertex graph has chromatic number ≥ (1/2 − ε)"n" / log₂ "n", and that its largest clique subdivision has at most "cn"^1/2 vertices for some constant "c". In this context it is worth noting that the probability also approaches one that a random "n"-vertex graph has Hadwiger number greater than or equal to its chromatic number, so the Hadwiger conjecture holds for random graphs with high probability; more precisely, the Hadwiger number is with high probability a constant times "n"/√log "n". [harvtxt|Bollobás|Catlin|Erdős|1980.]

Gerards and Seymour conjectured that every graph "G" with chromatic number "k" has a complete graph "K_k" as an "odd minor". Such a structure can be represented as a family of "k" vertex-disjoint subtrees of "G", each of which is two-colored, such that each pair of subtrees is connected by a monochromatic edge. Although graphs with no odd "K_k" minor are not necessarily sparse, a similar upper bound holds for them as it does for the standard Hadwiger conjecture: a graph with no odd "K_k" minor has chromatic number "χ"("G") = O(k √log "k"). [harvtxt|Geelen|Gerards|Reed|Seymour|2006; Kawarabayashi (to appear in "JCTB").]

By imposing more conditions on "G" than the number of colors it needs, it may be possible to prove the existence of larger minors than "K_k". One example of this is the snark theorem, that every cubic graph requiring four colors in any edge coloring has the Petersen graph as a minor, conjectured by W. T. Tutte and announced to be proved in 2001 by Robertson, Sanders, Seymour, and Thomas. [citation|last=Pegg|first=Ed, Jr.|authorlink=Ed Pegg, Jr.|title=Book Review: The Colossal Book of Mathematics|journal=Notices of the American Mathematical Society|volume=49|issue=9|year=2002|pages=1084–1086|url=http://www.ams.org/notices/200209/rev-pegg.pdf.]

Notes

References

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