Hadwiger conjecture (graph theory)
In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies It is known to be true for The conjecture is a generalization of the four color theorem and is considered to be one of the most important and challenging open problems in the field.
Reversing the implication, the conjecture can equivalently be stated in the following form. According to it, if all proper colorings of an undirected graph use or more colors, then one can find disjoint connected subgraphs of 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 vertex produces a complete graph on vertices as a minor
The conjecture was made by Hugo Hadwiger in 1943. call it "one of the deepest unsolved problems in graph theory".
Equivalent forms
One form of the Hadwiger conjecture is that, if there is no sequence of edge contractions that brings a graph to the complete then must have a vertex coloring with colors. Equivalently, if a given graph has no such coloring, then there is a way of contracting edges to produce as a graph minor.In a minimal of any contracting each color class of the coloring to a single vertex will produce a complete However, this contraction process does not produce a minor because there is no edge between any two vertices in the same color class, thus the contraction is not an edge contraction. Hadwiger's conjecture states that there exists a different way of properly edge contracting sets of vertices to single vertices, producing a complete in such a way that all the contracted sets are connected.
If denotes the family of graphs having the property that all minors of graphs in can be then it follows from the Robertson–Seymour theorem that can be characterized by a finite set of forbidden minors. Hadwiger's conjecture is that this set consists of a single forbidden
The Hadwiger number of a graph is the size of the largest complete graph that is a minor of (or equivalently can be obtained by contracting edges It is also known as the contraction clique number The Hadwiger conjecture can be stated in the simple algebraic form where denotes the chromatic number
Special cases and partial results
The case is trivial: a graph requires more than one color if and only if it has an edge, and that edge is itself a minor. The case 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 minor.In the same paper in which he introduced the conjecture, Hadwiger proved its truth The graphs with no 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 implies the four color theorem: for, if the conjecture is true, every graph requiring five or more colors would have a minor and would be nonplanar.
Klaus Wagner proved in 1937 that the case 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 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 -minor-free graph follows from the 4-colorability of each of the planar pieces.
proved the conjecture also using the four color theorem; their paper with this proof won the 1994 Fulkerson Prize. It follows from their proof that linklessly embeddable graphs, a three-dimensional analogue of planar graphs, have chromatic number at most five. Due to this result, the conjecture is known to be true but it remains unsolved for
For, some partial results are known: every 7-chromatic graph must contain either a minor or both a minor and a minor.
Every graph has a vertex with at most incident edges, from which it follows that a greedy coloring algorithm that removes this low-degree vertex, colors the remaining graph, and then adds back the removed vertex and colors it, will color the given graph with colors.
In the 1980s, Alexander V. Kostochka and Andrew Thomason both independently proved that every graph with no minor has average degree and can thus be colored using colors.
A sequence of improvements to this bound have led to a proof of -colorability for graphs without
Generalizations
György Hajós conjectured that Hadwiger's conjecture could be strengthened to subdivisions rather than minors: that is, that every graph with chromatic number contains a subdivision of a complete Hajós' conjecture is true but found counterexamples to this strengthened conjecture the cases and remain observed that Hajós' conjecture fails badly for random graphs: for in the limit as the number of vertices, goes to infinity, the probability approaches one that a random graph has chromatic and that its largest clique subdivision has vertices. In this context, it is worth noting that the probability also approaches one that a random 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 proportionalasked whether Hadwiger's conjecture could be extended to list coloring. every graph with list chromatic number has a clique minor. However, the maximum list chromatic number of planar graphs is 5, not 4, so the extension fails already for graphs. More generally, for there exist graphs whose Hadwiger number is and whose list chromatic number
Gerards and Seymour conjectured that every graph with chromatic number has a complete graph as an odd minor. Such a structure can be represented as a family of vertex-disjoint subtrees of, each of which is two-colored, such that each pair of subtrees is connected by a monochromatic edge. Although graphs with no odd 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 minor has chromatic number A 2025 preprint claims a lower bound of, disproving this conjecture.
By imposing extra conditions on, it may be possible to prove the existence of larger minors One example 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.