Triangle-free graph

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In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.

By Turán's theorem, the n-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible.

Triangle finding problem[edit]

The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph.

It is possible to test whether a graph with m edges is triangle-free in time O(m1.41).[1] Another approach is to find the trace of A3, where A is the adjacency matrix of the graph. The trace is zero if and only if the graph is triangle-free. For dense graphs, it is more efficient to use this simple algorithm which relies on matrix multiplication, since it gets the time complexity down to O(n2.373), where n is the number of vertices.

As Imrich, Klavžar & Mulder (1999) show, triangle-free graph recognition is equivalent in complexity to median graph recognition; however, the current best algorithms for median graph recognition use triangle detection as a subroutine rather than vice versa.

The decision tree complexity or query complexity of the problem, where the queries are to an oracle which stores the adjacency matrix of a graph, is Θ(n2). However, for quantum algorithms, the best known lower bound is Ω(n), but the best known algorithm is O(n1.29) due to Belovs (2011).

Independence number and Ramsey theory[edit]

An independent set of √n vertices in an n-vertex triangle-free graph is easy to find: either there is a vertex with greater than √n neighbors (in which case those neighbors are an independent set) or all vertices have fewer than √n neighbors (in which case any maximal independent set must have at least √n vertices).[2] This bound can be tightened slightly: in every triangle-free graph there exists an independent set of \Omega(\sqrt{n\log n}) vertices, and in some triangle-free graphs every independent set has O(\sqrt{n\log n}) vertices.[3] One way to generate triangle-free graphs in which all independent sets are small is the triangle-free process[4] in which one generates a maximal triangle-free graph by repeatedly adding randomly chosen edges that do not complete a triangle. With high probability, this process produces a graph with independence number O(\sqrt{n\log n}). It is also possible to find regular graphs with the same properties.[5]

These results may also be interpreted as giving asymptotic bounds on the Ramsey numbers R(3,t) of the form \Theta(\tfrac{t^2}{\log t}): if the edges of a complete graph on \Omega(\tfrac{t^2}{\log t}) vertices are colored red and blue, then either the red graph contains a triangle or, if it is triangle-free, then it must have an independent set of size t corresponding to a clique of the same size in the blue graph.

Coloring triangle-free graphs[edit]

Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch's theorem states that every triangle-free planar graph may be 3-colored.[6] However, nonplanar triangle-free graphs may require many more than three colors.

Mycielski (1955) defined a construction, now called the Mycielskian, for forming a new triangle-free graph from another triangle-free graph. If a graph has chromatic number k, its Mycielskian has chromatic number k + 1, so this construction may be used to show that arbitrarily large numbers of colors may be needed to color nonplanar triangle-free graphs. In particular the Grötzsch graph, an 11-vertex graph formed by repeated application of Mycielski's construction, is a triangle-free graph that cannot be colored with fewer than four colors, and is the smallest graph with this property.[7] Gimbel &Thomassen & (2000) and Nilli (2000) showed that the number of colors needed to color any m-edge triangle-free graph is

O \left(\frac{m^{1/3}}{(\log m)^{2/3}} \right)

and that there exist triangle-free graphs that have chromatic numbers proportional to this bound.

There have also been several results relating coloring to minimum degree in triangle-free graphs. Andrásfai, Erdős & Sós (1974) proved that any n-vertex triangle-free graph in which each vertex has more than 2n/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex. Motivated by this result, Erdős & Simonovits (1973) conjectured that any n-vertex triangle-free graph in which each vertex has at least n/3 neighbors can be colored with only three colors; however, Häggkvist (1981) disproved this conjecture by finding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size. Jin (1995) showed that any n-vertex triangle-free graph in which each vertex has more than 10n/29 neighbors must be 3-colorable; this is the best possible result of this type, because Häggkvist's graph requires four colors and has exactly 10n/29 neighbors per vertex. Finally, Brandt & Thomassé (2006) proved that any n-vertex triangle-free graph in which each vertex has more than n/3 neighbors must be 4-colorable. Additional results of this type are not possible, as Hajnal[8] found examples of triangle-free graphs with arbitrarily large chromatic number and minimum degree (1/3 − ε)n for any ε > 0.

See also[edit]

  • Monochromatic triangle problem, the problem of partitioning the edges of a given graph into two triangle-free graphs

References[edit]

Notes
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External links[edit]