10 Problems for Partitions of Triangle-free Graphs

Abstract

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős' Sparse Half Conjecture.Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition16. This result is sharp since the complete bipartite graph with class sizes 3n/4 and n/4 achieves equality, when n is a multiple of 4.Additionally, we discuss similar problems for K 4 -free graphs.