Fibre factorization and automorphisms of direct products of bipartite graphs
Abstract
We determine the automorphism groups of direct products of finite connected twin-free bipartite graphs whose factors are different from K2. The product is generally disconnected, so the problem requires both the automorphism groups of its components and a criterion for component isomorphism. We retain the order of the bipartition classes and use the fibre product over K2. An edge-relation construction, together with McKenzie's strict refinement theorem, yields unique fibre-prime factorization and a coordinate description of every specified isomorphism. After refining the original factors, component isomorphism classes are described by intersections of factor-symmetry orbits with a set of admissible labels. The stabilizer of a label acts faithfully as the fullautomorphism group of its component. These data determine the full automorphism group, including products with repeated or fibre-composite factors. For products of fibre powers of one prime graph, we compute the component groups and express their multiplicities through a subset-sum polynomial. We also obtain an explicit semidirect product for a mixed product with one nonbipartite factor and explain its relation to known stability results. An application recovers the known double-cover groups of bipartite generalized Petersen graphs.
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BibTeX
@misc{wang2026fibre,
title = {{Fibre factorization and automorphisms of direct products of bipartite graphs}},
author = {Xiaomeng Wang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00775},
eprint = {2611.00775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00775},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fibre factorization and automorphisms of direct products of bipartite graphs
AU - Xiaomeng Wang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00775
UR - https://arxiv.org/abs/2611.00775
AB - We determine the automorphism groups of direct products of finite connected twin-free bipartite graphs whose factors are different from K_2. The product is generally disconnected, so the problem requires both the automorphism groups of its components and a criterion for component isomorphism. We retain the order of the bipartition classes and use the fibre product over K_2. An edge-relation construction, together with McKenzie's strict refinement theorem, yields unique fibre-prime factorization and a coordinate description of every specified isomorphism. After refining the original factors, component isomorphism classes are described by intersections of factor-symmetry orbits with a set of admissible labels. The stabilizer of a label acts faithfully as the fullautomorphism group of its component. These data determine the full automorphism group, including products with repeated or fibre-composite factors. For products of fibre powers of one prime graph, we compute the component groups and express their multiplicities through a subset-sum polynomial. We also obtain an explicit semidirect product for a mixed product with one nonbipartite factor and explain its relation to known stability results. An application recovers the known double-cover groups of bipartite generalized Petersen graphs.
ER -