Adjacent vertices of small degree in minimal matching covered graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918421673803776 |
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| author | He, Xiaoling Lu, Fuliang Zhang, Heping |
| author_facet | He, Xiaoling Lu, Fuliang Zhang, Heping |
| contents | A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An edge is called a $k$-line if both of its end vertices are of degree $k$. Lovász and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] proved that a minimal matching covered bipartite graph different from $K_2$ has minimum degree 2 and contains at least $[(|V(G)|+15)/6]$ 2-lines by ear decompositions. He et al. [J. Graph Theory 111 (2026) 5--16] showed that the minimum degree of a minimal matching covered graph different from $K_2$ is either 2 or 3. In this paper, we prove that every minimal matching covered graph with at least 4 vertices contains at least two nonadjacent edges, each of which is either a 2-line or a 3-line. Consequently, we show that every minimal matching covered graph with at least 4 vertices and minimum degree 3 contains at least 4 vertices of degree 3. Furthermore, the lower bounds for both the number of 3-lines and the number of cubic vertices are sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00361 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Adjacent vertices of small degree in minimal matching covered graphs He, Xiaoling Lu, Fuliang Zhang, Heping Combinatorics A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An edge is called a $k$-line if both of its end vertices are of degree $k$. Lovász and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] proved that a minimal matching covered bipartite graph different from $K_2$ has minimum degree 2 and contains at least $[(|V(G)|+15)/6]$ 2-lines by ear decompositions. He et al. [J. Graph Theory 111 (2026) 5--16] showed that the minimum degree of a minimal matching covered graph different from $K_2$ is either 2 or 3. In this paper, we prove that every minimal matching covered graph with at least 4 vertices contains at least two nonadjacent edges, each of which is either a 2-line or a 3-line. Consequently, we show that every minimal matching covered graph with at least 4 vertices and minimum degree 3 contains at least 4 vertices of degree 3. Furthermore, the lower bounds for both the number of 3-lines and the number of cubic vertices are sharp. |
| title | Adjacent vertices of small degree in minimal matching covered graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.00361 |