Recursive constructions of amoebas

Abstract

Global amoebas are a wide and rich family of graphs that emerged from the study of certain Ramsey-Turán problems in 2-colorings of the edges of the complete graph Kn that deal with the appearance of unavoidable patterns once a certain amount of edges in each color is guaranteed. Indeed, it turns out that, as soon as such coloring constraints are satisfied and if n is sufficiently large, then every global amoeba can be found embedded in Kn such that it has half its edges in each color. Even more surprising, every bipartite global amoeba G is unavoidable in every tonal-variation, meaning that, for any pair of integers r, b such that r + b is the number of edges of G, there is a subgraph of Kn isomorphic to G with r edges in the first color and b edges in the second. The feature that makes global amoebas work are one-by-one edge replacements that leave the structure of the graph invariant. By means of a group theoretical approach, the dynamics of this feature can be modeled. As a counterpart to the global amoebas that “live” inside a possibly large complete graph Kn, we also consider local amoebas which are spanning subgraphs of Kn with the same feature.In an effort to highlight their richness and versatility, we present here three different recursive constructions of amoebas, two of them yielding interesting families per se and one of them offering a wide range of possibilities.