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Homomorphic Preimages of Geometric Cycles

Sally Cockburn

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.

Homomorphic Preimages of Geometric Cycles

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.

Paper Structure

This paper contains 6 sections, 3 theorems, 4 equations, 9 figures.

Table of Contents

  1. Basic Definitions
  2. Edge-Crossing Graph and Thickness Edge Colorings
  3. Easy Cases: $n=3$ and $n = 4$
  4. Harder Case: $n=5$
  5. 5-geocolorability
  6. Future Work

Key Result

Theorem 3.1

BC A geometric graph $\overline{G}$ is homomorphic to $\widehat{C}_4$ if and only if

Figures (9)

  • Figure 1: $\widehat{C}_4 \not \to \overline{K_2}$ and $\widehat{C}_5 \not \to \overline{C_3}$
  • Figure 2: $\overline{C}_4 \to \widehat{C}_4$
  • Figure 3: Homomorphism poset $\mathcal{C}_5$
  • Figure 4: $\widehat{C}_5$ with vertex and edge labels
  • Figure 5: $G \to C_5$ and $EX(\overline{G}) \to C_5$, but $\overline{G} \not \to \widehat{C}_5$
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Theorem 3.1
  • ...and 3 more