Cubic bricks that every b-invariant edge is forcing
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910716539174912 |
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| author | Zhang, Yaxian Lu, Fuliang Zhang, Heping |
| author_facet | Zhang, Yaxian Lu, Fuliang Zhang, Heping |
| contents | A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges. Denote by b(G) the number of bricks of G. An edge e of G is removable if G-e is also matching covered, and solitary (or forcing) if after the removal of the two end vertices of e, the left graph has a unique perfect matching. Furthermore, a removable edge e of a brick G is b-invariant if b(G-e) = 1.
Lucchesi and Murty proposed a problem of characterizing bricks, distinct from K4, the prism and the Petersen graph, in which every b-invariant edge is forcing. We answer the problem for cubic bricks by showing that there are exactly ten cubic bricks, including K4, the prism and the Petersen graph, every b-invariant edge of which is forcing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cubic bricks that every b-invariant edge is forcing Zhang, Yaxian Lu, Fuliang Zhang, Heping Combinatorics A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges. Denote by b(G) the number of bricks of G. An edge e of G is removable if G-e is also matching covered, and solitary (or forcing) if after the removal of the two end vertices of e, the left graph has a unique perfect matching. Furthermore, a removable edge e of a brick G is b-invariant if b(G-e) = 1. Lucchesi and Murty proposed a problem of characterizing bricks, distinct from K4, the prism and the Petersen graph, in which every b-invariant edge is forcing. We answer the problem for cubic bricks by showing that there are exactly ten cubic bricks, including K4, the prism and the Petersen graph, every b-invariant edge of which is forcing. |
| title | Cubic bricks that every b-invariant edge is forcing |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.17295 |