Universality of Graphs with Few Triangles and Anti-Triangles

On Tensor Product of Graphs, Girth and Triangles

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

Triangles in random graphs

Triangles in Random Graphs

OF THE DISSERTATION Triangles in Random Graphs by Robert DeMarco Dissertation Director: Jeff Kahn We prove four separate results. These results will appear or have appeared in various papers (see [10], [11], [12], [13]). For a gentler introduction to these results, the reader is directed to the first chapter of this thesis. Let G = Gn,p. With ξk = ξ n,p k the number of copies of Kk in G, p ≥ n−...

Planar 4-critical graphs with four triangles

By the Grünbaum-Aksenov Theorem (extending Grötzsch’s Theorem) every planar graph with at most three triangles is 3-colorable. However, there are infinitely many planar 4-critical graphs with exactly four triangles. We describe all such graphs. This answers a question of Erdős from 1990.

Counting Triangles in Massive Graphs with MapReduce

Graphs and networks are used to model interactions in a variety of contexts. There is a growing need to quickly assess the characteristics of a graph in order to understand its underlying structure. Some of the most useful metrics are triangle-based and give a measure of the connectedness of mutual friends. This is often summarized in terms of clustering coefficients, which measure the likeliho...