Graphs isomorphisms under edge-replacements and the family of amoebas

TL;DR

This paper systematically studies amoebas, a family of graphs with unique edge-replacement properties, revealing their structure, diversity, and algebraic characteristics, and introduces constructions demonstrating their complexity and richness.

Contribution

It introduces the concept of amoebas, characterizes their structure via algebraic groups, and provides constructions showing their diversity and complexity.

Findings

Any connected graph can be a component of a global amoeba

Global amoebas can be very dense with large clique and chromatic numbers

Constructed global amoeba trees with Fibonacci-like structure and large maximum degree

Abstract

This paper offers a systematic study of a family of graphs called amoebas. Amoebas recently emerged from the study of forced patterns in $2$-colorings of the edges of the complete graph in the context of Ramsey-Turan theory and played an important role in extremal zero-sum problems. Amoebas are graphs with a unique behavior with regards to the following operation: Let $G$ be a graph and let $e\in E(G)$ and $e'\in E(\overline{G})$. If the graph $G'=G-e+e'$ is isomorphic to $G$, we say $G'$ is obtained from $G$ by performing a \emph{feasible edge-replacement}. We call $G$ a \emph{local amoeba} if, for any two copies $G_1$, $G_2$ of $G$ on the same vertex set, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called \emph{global amoeba} if there is an integer $t_0 \ge 0$ such that $G \cup tK_1$ is a local amoeba for all $t \ge t_0$. To model the dynamics of the feasible edge-replacements of $G$, we define a group ${\rm Fer}(G)$ that satisfies that $G$ is a local amoeba if and only if ${\rm Fer}(G) \cong S_n$, where $n$ is the order of $G$. Via this algebraic setting, a deeper understanding of the structure of amoebas and their intrinsic properties comes into light. Moreover, we present different constructions that prove the richness of these graph families showing, among other things, that any connected graph can be a connected component of a global amoeba, that global amoebas can be very dense and that they can have, in proportion to their order, large clique and chromatic numbers. Also, a family of global amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is constructed.