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The original motivation for the study of triangle removal lemma was the Ruzsa–Szemerédi problem.Its initial formulation due to Imre Z. Ruzsa and Szemerédi from 1978 was slightly weaker than the triangle removal lemma used nowadays and can be roughly stated as follows: every locally linear graph on vertices contains () edges. [6]
To convert a graph with a unique triangle per edge into a triple system, let the triples be the triangles of the graph. No six points can include three triangles without either two of the three triangles sharing an edge or all three triangles forming a fourth triangle that shares an edge with each of them.
A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles. The line graph of the complete graph K n is also known as the triangular graph, the Johnson graph J(n, 2), or the complement of the Kneser graph ...
The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (vertices) during one month in summer 2013. [6] Graphs can be used to model many types of relations and processes in physical, biological, [7] [8] social and information systems. [9] Many practical problems can be represented by graphs.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
But if not, each must be oppositely coloured, completing triangle xyz of that colour. A 2-edge-labeling of K 5 with no monochromatic K 3. Suppose the edges of a complete graph on 6 vertices are coloured red and blue. Pick a vertex, v. There are 5 edges incident to v and so (by the pigeonhole principle) at least 3 of them must be the same colour.
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 directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
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