On the minimum degree of minimal $k$-$\{1,2\}$-factor critical $k$-planar graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914384807198720 |
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| author | Pereyra, Kevin |
| author_facet | Pereyra, Kevin |
| contents | A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical for any edge $e$ in $G$. In 1998, Favaron and Shi posed the conjecture that every minimal $k$-factor-critical graph is of minimum degree $k+1$.
A natural extension of this notion arises from $\{1,2\}$-factors. A spanning subgraph of $G$ is called a $\{1,2\}$-factor if each of its components is a regular graph of degree one or two. A graph is $k$-\emph{$\{1,2\}$-factor critical} if the removal of any $k$ vertices results in a graph with a $\{1,2\}$-factor.
A recent conjecture in the area states that every minimal $k$-$\{1,2\}$-factor critical graph $G$ satisfies $k+1\le δ(G)\le k+2$. In this paper, we prove that the conjecture holds for $k$-planar graphs, that is, graphs in which the deletion of any set of $k$ vertices yields a planar graph. In particular, this resolves the conjecture for planar graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10317 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the minimum degree of minimal $k$-$\{1,2\}$-factor critical $k$-planar graphs Pereyra, Kevin Combinatorics A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical for any edge $e$ in $G$. In 1998, Favaron and Shi posed the conjecture that every minimal $k$-factor-critical graph is of minimum degree $k+1$. A natural extension of this notion arises from $\{1,2\}$-factors. A spanning subgraph of $G$ is called a $\{1,2\}$-factor if each of its components is a regular graph of degree one or two. A graph is $k$-\emph{$\{1,2\}$-factor critical} if the removal of any $k$ vertices results in a graph with a $\{1,2\}$-factor. A recent conjecture in the area states that every minimal $k$-$\{1,2\}$-factor critical graph $G$ satisfies $k+1\le δ(G)\le k+2$. In this paper, we prove that the conjecture holds for $k$-planar graphs, that is, graphs in which the deletion of any set of $k$ vertices yields a planar graph. In particular, this resolves the conjecture for planar graphs. |
| title | On the minimum degree of minimal $k$-$\{1,2\}$-factor critical $k$-planar graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.10317 |