Can I read Unavoidable Patterns in $2$-Colorings of the Complete Bipartite Graph on EtoBox?

Unavoidable Patterns in $2$-Colorings of the Complete Bipartite Graph by Hansberg, Adriana; Ventura, Denae is a scholarly article available to read on EtoBox.

What is Unavoidable Patterns in $2$-Colorings of the Complete Bipartite Graph about?

We determine the colored patterns that appear in any $2$-edge coloring of $K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We prove the existence of a positive integer $z_2$ such that any $2$-edge coloring of $K_{n,n}$ with at least $z_2$ edges in each color contains at least one of these patterns. We give a general upper bound for $z_2$ and prove its tightness for some cases. We define the concepts of bipartite $r$-tonality and bipartite omnitonality using the complete bipartite graph as a base graph. We provide a characterization for bipartite $r$-tonal graphs and prove that every tree is bipartite omnitonal. Finally, we define the bipartite balancing number and provide the exact bipartite balancing number for paths and stars.

Author

Hansberg, Adriana; Ventura, Denae

Published

2024

Language

EN