2024/07/11 by Adriana Hansberg, Hansberg, Adriana, Denae Ventura +1
Computer Science · Engineering · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2407.08873
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine the colored patterns that appear in any 2-edge coloring of Kn,n, with n large enough and with sufficient edges in each color. We prove the existence of a positive integer z2 such that any 2-edge coloring of Kn,n with at least z2 edges in each color contains at least one of these patterns. We give a general upper bound for z2 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.