# Triangle-free graph

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.

## Contents

## Triangle finding problem

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(*m*^{1.41}).^{1} Another approach is to find the trace of *A*^{3}, 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(*n*^{2.373}), 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(*n*^{2}). However, for quantum algorithms, the best known lower bound is Ω(*n*), but the best known algorithm is O(*n*^{1.29}) due to Belovs (2011).

## Independence number and Ramsey theory

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 vertices, and in some triangle-free graphs every independent set has 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 . 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 : if the edges of a complete graph on 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

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

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 2*n*/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2*n*/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 10*n*/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 10*n*/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

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

## References

- Notes

**^**Alon, Yuster & Zwick (1994).**^**Boppana & Halldórsson (1992) p. 184, based on an idea from an earlier coloring approximation algorithm of Avi Wigderson.**^**Kim (1995).**^**Erdős, Suen & Winkler (1995); Bohman (2008)**^**Alon, Ben-Shimon & Krivelevich (2008).**^**Grötzsch (1959); Thomassen (1994)).**^**Chvátal (1974).**^**see Erdős & Simonovits (1973).

- Sources

- Alon, N.; Ben-Shimon, S.; Krivelevich, M. (2008). "A note on regular Ramsey graphs". arXiv:0812.2386..
- Alon, N.; Yuster, R.; Zwick, U. (1994), "Finding and counting given length cycles",
*Proceedings of the 2nd European Symposium on Algorithms, Utrecht, The Netherlands*, pp. 354–364. - Andrásfai, B.; Erdős, P.; Sós, V. T. (1974), "On the connection between chromatic number, maximal clique and minimal degree of a graph",
*Discrete Mathematics***8**(3): 205–218, doi:10.1016/0012-365X(74)90133-2. - Bohman, T. (2008). "The triangle-free process". arXiv:0806.4375..
- Boppana, Ravi; Halldórsson, Magnús M. (1992), "Approximating maximum independent sets by excluding subgraphs",
*BIT***32**(2): 180–196, doi:10.1007/BF01994876, MR 1172185. - Brandt, S.; Thomassé, S. (2006),
*Dense triangle-free graphs are four-colorable: a solution to the Erdős-Simonovits problem*. - Chiba, N.; Nishizeki, T. (1985), "Arboricity and subgraph listing algorithms",
*SIAM Journal on Computing***14**(1): 210–223, doi:10.1137/0214017. - Chvátal, Vašek (1974), "The minimality of the Mycielski graph",
*Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973)*, Lecture Notes in Mathematics**406**, Springer-Verlag, pp. 243–246. - Erdős, P.; Simonovits, M. (1973), "On a valence problem in extremal graph theory",
*Discrete Mathematics***5**(4): 323–334, doi:10.1016/0012-365X(73)90126-X. - Erdős, P.; Suen, S.; Winkler, P. (1995), "On the size of a random maximal graph",
*Random Structures and Algorithms***6**(2–3): 309–318, doi:10.1002/rsa.3240060217. - Gimbel, John; Thomassen, Carsten (2000), "Coloring triangle-free graphs with fixed size",
*Discrete Mathematics***219**(1-3): 275–277, doi:10.1016/S0012-365X(00)00087-X. - Grötzsch, H. (1959), "Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel",
*Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe***8**: 109–120. - Häggkvist, R. (1981), "Odd cycles of specified length in nonbipartite graphs",
*Graph Theory (Cambridge, 1981)*, pp. 89–99. - Imrich, Wilfried; Klavžar, Sandi; Mulder, Henry Martyn (1999), "Median graphs and triangle-free graphs",
*SIAM Journal on Discrete Mathematics***12**(1): 111–118, doi:10.1137/S0895480197323494, MR 1666073. - Itai, A.; Rodeh, M. (1978), "Finding a minimum circuit in a graph",
*SIAM Journal on Computing***7**(4): 413–423, doi:10.1137/0207033. - Jin, G. (1995), "Triangle-free four-chromatic graphs",
*Discrete Mathematics***145**(1-3): 151–170, doi:10.1016/0012-365X(94)00063-O. - Kim, J. H. (1995), "The Ramsey number has order of magnitude ",
*Random Structures and Algorithms*(3 ed.)**7**: 173–207. - Magniez, Frederic; Santha, Miklos; Szegedy, Mario (2003). "Quantum Algorithms for the Triangle Problem". arXiv:quant-ph/0310134..
- Mycielski, J. (1955), "Sur le coloriage des graphes",
*Colloq. Math.***3**: 161–162. - Nilli, A. (2000), "Triangle-free graphs with large chromatic numbers",
*Discrete Mathematics***211**(1–3): 261–262, doi:10.1016/S0012-365X(99)00109-0. - Shearer, J. B. (1983), "Note on the independence number of triangle-free graphs",
*Discrete Mathematics***46**(1): 83–87, doi:10.1016/0012-365X(83)90273-X. - Thomassen, C. (1994), "Grötzsch's 3-color theorem",
*Journal of Combinatorial Theory, Series B***62**(2): 268–279, doi:10.1006/jctb.1994.1069. - Belovs, Aleksandrs (2011). "Span Programs for Functions with Constant-Sized 1-certificates". arXiv:1105.4024..

## External links

HPTS - Area Progetti - Edu-Soft - JavaEdu - N.Saperi - Ass.Scuola.. - TS BCTV - TS VideoRes - TSODP - TRTWE | ||