Wikipedia / Interval coloring
Wikipedia / Interval coloring
In graph theory, the interval chromatic number of an ordered graph can be partitioned.
Definition and basic properties
An ordered graph together with a specified linear ordering, an interval coloring into independent sets of consecutive vertices (called intervals or parts). The interval chromatic number.
Since the parts of an interval coloring are independent sets:
,
is the ordinary chromatic number.
An ordered graph is called interval k-chromatic (or k-ichromatic.
Computational complexity
It is of particular interest that the interval chromatic number is easily computable. By a simple greedy algorithm independent intervals. This is in sharp contrast with computing the usual chromatic number of a graph, where even finding an approximation is an NP hard task.
and its isomorphic graphs, the chromatic number remains the same, but the interval chromatic number may differ depending on the ordering of the vertex set.
Extremal theory
Pach and Tardos proved a fundamental theorem relating the interval chromatic number to extremal graph theory:
vertices can have is:
of forbidden ordered subgraphs with:
Relation to forbidden patterns
The interval chromatic number plays a crucial role in forbidden pattern problems for ordered graphs and zero-one matrices. For a 2-ichromatic ordered graph, the extremal problem becomes particularly interesting, as the quadratic term vanishes in the extremal function formula..
The forbidden pattern problems connect to several problems in discrete geometry, including the unit distance graph problem, the planar segment-center problem, and the finding of Davenport–Schinzel sequences.