10 Problems for Partitions of Trianglefree Graphs
Abstract
We will state 10 problems, and solve some of them, for partitions in trianglefree 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 trianglefree graph on $n$ vertices has a partition $V(G)=A\cup B$ with $A=B=n/2$ such that $e(G[A])+e(G[B])\leq n^2/16$. 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.15764
 arXiv:
 arXiv:2203.15764
 Bibcode:
 2022arXiv220315764B
 Keywords:

 Mathematics  Combinatorics