2022/06/21 by David E. Roberson, Roberson, David E. · 1 voice · 4 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2206.10321
openalex publication_date 2022/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce (weak) oddomorphisms of graphs which are homomorphisms with additional constraints based on parity. These maps turn out to have interesting properties (e.g., they preserve planarity), particularly in relation to homomorphism indistinguishability. Graphs G and H are *homomorphism indistinguishable* over a family F if \hom(F,G) = \hom(F,H) for all F ∈ F, where \hom(F,G) is the number of homomorphisms from F to G. A classical result of Lovász says that isomorphism is equivalent to homomorphism indistinguishability over the class of all graphs. In recent years it has been shown that many homomorphism indistinguishability relations have natural algebraic and/or logical formulations. Currently, much research in this area is focused on finding such reformulations. We aim to broaden the scope of current research on homomorphism indistinguishability by introducing new concepts/constructions and proposing several conjectures/questions. In particular, we conjecture that every family closed under disjoint unions and minors gives rise to a distinct homomorphism indistinguishability relation. We also show that if F is a family of graphs closed under disjoint unions, restrictions to connected components, and weak oddomorphisms, then F satisfies a certain maximality or closure property: homomorphism indistinguishability over F of G and H does not imply \hom(F,G) = \hom(F,H) for any F ∉ F. This allows us to answer a question raised over ten years ago, showing that homomorphism indistinguishability over graphs of bounded degree is not equivalent to isomorphism.