Much research about triangle-free graphs has focused on graph coloring.

However, nonplanar triangle-free graphs may require many more than three colors.

There have also been several results relating coloring to minimum degree in triangle-free graphs.

The complement of any triangle-free graph is a claw-free graph.

A triangle-free graph is a graph with no induced cycle of length three.

Finally, proved that any n-vertex triangle-free graph in which each vertex has more than n/3 neighbors must be 4-colorable.

A triangle-free graph is outerplanar if and only if it does not contain a subdivision of K.

The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle.

A triangle-free graph is a graph that has no cliques other than its vertices and edges.

Mycielski's theorem that there exist triangle-free graphs with arbitrarily large chromatic number.

For example the triangle-free graphs are the graphs that do not have a triangle graph as an induced subgraph.

It is a triangle-free graph with minimum degree four, so it cannot be changed by any YΔY-reduction.

In particular, it is not possible for a Halin graph to be a triangle-free graph nor a bipartite graph.

For example, the largest triangle-free graph on '2n' vertices is a complete bipartite graph 'Kn,n'.

Unlike later graphs in this sequence, the Grötzsch graph is the smallest triangle-free graph with its chromatic number .

Similarly, used the construction, starting with the Grötzsch graph, to generate many 4-critical triangle-free graphs, which they showed to be difficult to color using traditional backtracking algorithms.

Hereditary maximal-clique irreducible graphs include triangle-free graphs, bipartite graphs, and interval graphs.

Every minor-closed property is monotone, but not necessarily vice versa; for instance, minors of triangle-free graphs are not necessarily themselves triangle-free.

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.

Graph families that have larger numbers of graphs than this, such as the bipartite graphs or the triangle-free graphs, do not have adjacency labeling schemes.

Therefore, a minimal counterexample to the cycle double cover conjecture must be a triangle-free graph, ruling out some snarks such as Tietze's graph which contain triangles.

In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5.

The theorem cannot be generalized to all nonplanar triangle-free graphs: not every nonplanar triangle-free graph is 3-colorable.

This connection between Latin transversals and rainbow matchings in K has inspired additional interest in the study of rainbow matchings in triangle-free graphs.

For triangle-free graphs, or more generally graphs in which the neighborhood of every vertex is sufficiently sparse, O(Δ/log Δ) colors suffice.