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arXiv · 1507.02758

Homomorphic Preimages of Geometric Cycles

Sally Cockburn
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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.

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Sally Cockburn. 2015-07-10. Homomorphic Preimages of Geometric Cycles. https://doi.org/10.7151/dmgt.2025

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