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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.
A convenient and symmetric reformulation of the Erdős–Hajnal conjecture is that for every graph , the -free graphs necessarily contain a perfect induced subgraph of polynomial size. This is because every perfect graph necessarily has either a clique or independent set of size proportional to the square root of their number of vertices, and ...
A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.
In graph theory, a cograph, or complement-reducible graph, or P 4-free graph, is a graph that can be generated from the single-vertex graph K 1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K 1 and is closed under complementation and disjoint union.
The theorem cannot be generalized to all nonplanar triangle-free graphs: not every nonplanar triangle-free graph is 3-colorable. In particular, the Grötzsch graph and the Chvátal graph are triangle-free graphs requiring four colors, and the Mycielskian is a transformation of graphs that can be used to construct triangle-free graphs that ...
In graph theory, a branch of mathematics, a linear forest is a kind of forest where each component is a path graph, [1]: 200 or a disjoint union of nontrivial paths. [2]: 246 Equivalently, it is an acyclic and claw-free graph. [3]: 130, 131 An acyclic graph where every vertex has degree 0, 1, or 2 is a linear forest.
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