The Homomorphism Poset of $K_{3,3}$
Sally Cockburn

TL;DR
This paper characterizes the structure of the homomorphism poset of the complete bipartite graph K_{3,3} by analyzing geometric realizations and their homomorphisms in the plane.
Contribution
It determines the entire homomorphism poset of K_{3,3}, providing a detailed classification of geometric realizations and their relationships.
Findings
The homomorphism poset of K_{3,3} is fully characterized.
A classification of geometric realizations of K_{3,3} is provided.
The structure of the partial order induced by geometric homomorphisms is described.
Abstract
A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). Geometric homomorphisms can be used to define a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. In this paper, the homomorphism poset of K_{3,3} is determined.
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Analytic Number Theory Research
