Wheel-like bricks and minimal matching covered graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910979087925248 |
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| author | He, Xiaoling Lu, Fuliang Xue, Jinxin |
| author_facet | He, Xiaoling Lu, Fuliang Xue, Jinxin |
| contents | A connected graph G with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge e in a matching covered graph G is removable if G-e is matching covered. A pair {e; f} of edges of a matching covered graph G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A 3-connected graph is a brick if the removal of any two distinct vertices, the left graph has a perfect matching. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty proposed a problem of characterizing wheel-like bricks. We show that every wheel-like brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels in a certain manner. A matching covered graph is minimal if the removal of any edge, the left graph is not matching covered. Lovasz and Plummer proved that the minimum degree of a minimal matching covered bipartite graph different from K2 is 2 by ear decompositions in 1977. By the properties of wheel-like bricks, we prove that the minimum degree of a minimal matching covered graph other than K2 is 2 or 3. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wheel-like bricks and minimal matching covered graphs He, Xiaoling Lu, Fuliang Xue, Jinxin Combinatorics A connected graph G with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge e in a matching covered graph G is removable if G-e is matching covered. A pair {e; f} of edges of a matching covered graph G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A 3-connected graph is a brick if the removal of any two distinct vertices, the left graph has a perfect matching. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty proposed a problem of characterizing wheel-like bricks. We show that every wheel-like brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels in a certain manner. A matching covered graph is minimal if the removal of any edge, the left graph is not matching covered. Lovasz and Plummer proved that the minimum degree of a minimal matching covered bipartite graph different from K2 is 2 by ear decompositions in 1977. By the properties of wheel-like bricks, we prove that the minimum degree of a minimal matching covered graph other than K2 is 2 or 3. |
| title | Wheel-like bricks and minimal matching covered graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.16465 |