We prove that if is a subgraph of complete multipartite graph , then contains connected component satisfying . use this to every 3-coloring the edges monochromatic with at least edges. further show such coloring has circuit fraction This verifies conjecture Conlon and Tyomkyn. Moreover, for general we -coloring

The design is described of a parallel version of Tarjan’s algorithm for the determination of equivalence classes in graphs that represent images. Distribution of the vertices of the graph over a number of processes leads to a message passing algorithm. The algorithm is mapped to a shared-memory architecture by means of POSIX threads. It is applied to the determination of connected components in...

Abstract We study amenability of definable groups and topological groups, prove various results, briefly described below. Among our main technical tools, interest in its own right, is an elaboration on strengthening the Massicot-Wagner version (Massicot Wagner J Ec Polytech Math 2:55–63, 2015) stabilizer theorem (Hrushovski Am Soc 25:189–243, 2012), also some results about measures measure-like...

In the 1960’s, Tutte presented a decomposition of a 2-connected finite graph into 3-connected graphs, cycles and bonds. This decomposition has been used to reduce problems on 2-connected graphs to problems on 3-connected graphs. Motivated by a problem concerning accumulation points of infinite planar graphs, we generalize Tutte’s decomposition to include all infinite 2-connected graphs.