Homomorphic Preimages of Geometric Cycles

TL;DR

This paper investigates the conditions under which geometric graphs can be homomorphically mapped to cycles of length less than 6, extending classical graph coloring concepts to geometric settings.

Contribution

It provides necessary and sufficient conditions for a geometric graph to be homomorphically mapped to cycles C_n for n<6, in the context of geometric graph colorability.

Findings

Characterization of C_n-colorability for geometric graphs with n<6

Extension of classical graph homomorphism concepts to geometric graphs

Conditions for geometric homomorphisms preserving edge crossings

Abstract

A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism from G to H. A classic problem is to characterize the family of homomorphic preimages of a given graph H. A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. A geometric homomorphism (resp. isomorphism) is a graph homomorphism (resp. isomorphism) that preserves edge crossings (resp. and non-crossings). The homomorphism posetof a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph G is H-colorable if there is a geometric homomorphism from G to some element of the homomorphism poset of H. We provide necessary and sufficient conditions for a geometric graph to be C_n-colorable for n less than 6.