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New mixed cages, bounds and computational results
Abstract
An (r,z,g)-mixed cage is a mixed graph that is r-regular in edges, z-regular in arcs, of girth g and of minimum order n=n(r,z,g). We give a general lower bound for n(r,z,g). In the case g=3, we determine the exact value n(r,z,3)=r+2z+1. For g=4 we give two improved lower bounds and we determine the exact value n(r,z,4)=2r+4z for r≥ 2z−2. We also present computational results, obtaining mixed cages and computational lower bounds by means of exhaustive backtracking searches with symmetry reductions. Moreover, we give an upper bound for n(1,z,g) that matches all our computed values.
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BibTeX
@misc{cruz2026mixed,
title = {{New mixed cages, bounds and computational results}},
author = {Claudia Marlene De la Cruz and Miguel Ángel Pizaña},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14110},
eprint = {2610.14110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14110},
}
RIS
TY - UNPB
TI - New mixed cages, bounds and computational results
AU - Claudia Marlene De la Cruz
AU - Miguel Ángel Pizaña
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14110
UR - https://arxiv.org/abs/2610.14110
AB - An (r,z,g)-mixed cage is a mixed graph that is r-regular in edges, z-regular in arcs, of girth g and of minimum order n=n(r,z,g). We give a general lower bound for n(r,z,g). In the case g=3, we determine the exact value n(r,z,3)=r+2z+1. For g=4 we give two improved lower bounds and we determine the exact value n(r,z,4)=2r+4z for r≥ 2z−2. We also present computational results, obtaining mixed cages and computational lower bounds by means of exhaustive backtracking searches with symmetry reductions. Moreover, we give an upper bound for n(1,z,g) that matches all our computed values.
ER -