Albertson conjecture

In combinatorial mathematics, the Albertson conjecture is an unproven relationship between the crossing number and the chromatic number of a graph.

It is named after Michael O. Albertson, a professor at Smith College, who stated it as a conjecture in 2007;[1] it is one of his many conjectures in graph coloring theory.

[2] The conjecture states that, among all graphs requiring

colors, the complete graph

Equivalently, if a graph can be drawn with fewer crossings than

It is straightforward to show that graphs with bounded crossing number have bounded chromatic number: one may assign distinct colors to the endpoints of all crossing edges and then 4-color the remaining planar graph.

Albertson's conjecture replaces this qualitative relationship between crossing number and coloring by a more precise quantitative relationship.

Specifically, a different conjecture of Richard K. Guy (1972) states that the crossing number of the complete graph

is It is known how to draw complete graphs with this many crossings, by placing the vertices in two concentric circles; what is unknown is whether there exists a better drawing with fewer crossings.

Therefore, a strengthened formulation of the Albertson conjecture is that every

-chromatic graph has crossing number at least as large as the right hand side of this formula.

A weaker form of the conjecture, proven by M. Schaefer,[3] states that every graph with chromatic number

(using big omega notation), or equivalently that every graph with crossing number

Albertson, Cranston & Fox (2009) published a simple proof of these bounds, by combining the fact that every minimal

(because otherwise greedy coloring would use fewer colors) together with the crossing number inequality according to which every graph

Using the same reasoning, they show that a counterexample to Albertson's conjecture for the chromatic number

The Albertson conjecture is vacuously true for

has crossing number zero, so the conjecture states only that the

of Albertson's conjecture is equivalent to the four color theorem, that any planar graph can be colored with four or fewer colors, for the only graphs requiring fewer crossings than the one crossing of

are the planar graphs, and the conjecture implies that these should all be at most 4-chromatic.

Through the efforts of several groups of authors the conjecture is now known to hold for all

, Luiz and Richter presented a family of

[5] There is also a connection to the Hadwiger conjecture, an important open problem in combinatorics concerning the relationship between chromatic number and the existence of large cliques as minors in a graph.

[6] A variant of the Hadwiger conjecture, stated by György Hajós, is that every

; if this were true, the Albertson conjecture would follow, because the crossing number of the whole graph is at least as large as the crossing number of any of its subdivisions.

However, counterexamples to the Hajós conjecture are now known,[7] so this connection does not provide an avenue for proof of the Albertson conjecture.