Inversion diameter and 2-edge-colored homomorphisms

TL;DR

This paper investigates the inversion diameter of graphs, providing new bounds and relationships with graph properties like acyclic chromatic number, girth, and maximum degree, using homomorphism techniques.

Contribution

It introduces bounds on inversion diameter via homomorphism-universal graphs, improves existing bounds related to acyclic chromatic number, and explores inversion diameters for planar, girth-specific, and subdivided graphs.

Findings

Inversion diameter is linearly bounded by acyclic chromatic number.

Lower bound of 6 for planar graphs' inversion diameter.

For triangle-free graphs, diam(G) <= D + log D, supporting a conjecture.

Abstract

In an oriented graph, the inversion of a subset of vertices X is the operation reversing the direction of every arc with both endpoints in X. Given a graph G, the inversion distance between two orientations G is the minimum number of inversions transforming one into the other. The inversion diameter diam(G) is the maximum such distance over all pairs of orientations of G. Through an equivalent formulation of inversions over 2-edge-colorings of G, we introduce the use of homomorphism-universal 2-edge-colored graphs to obtain bounds on the inversion diameter of various classes of graphs. Our first result upper bounds the inversion diameter by a linear function of the acyclic chromatic number, improving on the previous quadratic dependency. We then consider the inversion diameter of planar graphs, exhibiting a lower bound of 6, as well as new lower and upper bounds for those of a given girth, in particular settling the girth 7 case. We then show that any triangle-free graph G with maximum degree D satisfies diam(G) <= D + log D, making progress on the conjecture of Havet et al. that diam(G) <= D. Finally, we prove a general result about subdivisions: if a graph has inversion diameter k, any of its subdivisions has inversion diameter at most k + log k + 5.