Cubic vertices in minimal braces
Abstract
Braces play a fundamental role in matching theory, as they, together with bricks, constitute the basic building blocks of matching covered graphs in the tight cut decomposition. A brace is minimal if deleting any edge from it results in a graph that is not a brace. We prove that every perfect matching of a minimal brace of order at least six contains an edge whose ends both have degree three. Consequently, every such brace has at least three edges with this property. Combining this result with the forest structure induced by the noncubic vertices, we show that every minimal brace G of order n≥6 and size m, other than K3,3, satisfies n3(G)≥max{8,⌈(2n+8)/5⌉, ⌈(m−n+4)/2⌉}, where n3(G) is the number of cubic vertices of G. Finally, we characterize the graphs attaining the constant lower bound: a minimal brace has exactly eight cubic vertices if and only if it is isomorphic to B8, B10, Q10+, or Q12.
Related
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BibTeX
@misc{zhang2026cubic,
title = {{Cubic vertices in minimal braces}},
author = {Yipei Zhang and Xiumei Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06933},
eprint = {2610.06933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06933},
}
RIS
TY - UNPB
TI - Cubic vertices in minimal braces
AU - Yipei Zhang
AU - Xiumei Wang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.06933
UR - https://arxiv.org/abs/2610.06933
AB - Braces play a fundamental role in matching theory, as they, together with bricks, constitute the basic building blocks of matching covered graphs in the tight cut decomposition. A brace is minimal if deleting any edge from it results in a graph that is not a brace. We prove that every perfect matching of a minimal brace of order at least six contains an edge whose ends both have degree three. Consequently, every such brace has at least three edges with this property. Combining this result with the forest structure induced by the noncubic vertices, we show that every minimal brace G of order n≥6 and size m, other than K_3,3, satisfies n_3(G)≥max{8,⌈(2n+8)/5⌉, ⌈(m−n+4)/2⌉}, where n_3(G) is the number of cubic vertices of G. Finally, we characterize the graphs attaining the constant lower bound: a minimal brace has exactly eight cubic vertices if and only if it is isomorphic to B_8, B_10, Q_10^+, or Q_12.
ER -