On the maximum partial-dual genus of a planar graph
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2025
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| _version_ | 1866908285005725696 |
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| author | Chen, Jiaying Jin, Xian'an Zhang, Gang |
| author_facet | Chen, Jiaying Jin, Xian'an Zhang, Gang |
| contents | Let $G$ be an embedded graph and $A$ an edge subset of $G$. The partial dual of $G$ with respect to $A$, denoted by $G^A$, can be viewed as the geometric dual $G^*$ of $G$ over $A$. If $A=E(G)$, then $G^A=G^*$. Denote by $γ(G^A)$ the genus of the embedded graph $G^A$. The maximum partial-dual genus of $G$ is defined as $$^\partialγ_{M}(G):=\max_{A \subseteq E(G)}γ(G^A).$$ For any planar graph $G$, it had been proved that $^\partialγ_{M}(G)$ does not rely on the embeddings of $G$. In this paper, we further prove that if $G$ is a connected planar graph of order $n\geq 2$, then $^{\partial}γ_{M}(G)\geq \frac{n-n_2-2n_1}{2}+1$, where $n_i$ is the number of vertices of degree $i$ in $G$. As a consequence, if $G$ is a connected planar graph of order $n$ with minimum degree at least 3, then $^{\partial}γ_{M}(G) \geq \frac{n}{2}+1$. Denote by $G^c$ the complement of a graph $G$ and by $χ(G^c)$ the chromatic number of $G^c$. Moreover, we prove that if $G \ncong K_4$ is a $λ$-edge-connected planar graph of order $n$, then $^{\partial}γ_{M}(G) \geq f(n,λ,χ(G^c))$, where $f(n,λ,χ(G^c))$ is a function of $n$, $λ$ and $χ(G^c)$. The first lower bound is tight for any $n$, and the second lower bound is tight for some 3-edge-connected graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximum partial-dual genus of a planar graph Chen, Jiaying Jin, Xian'an Zhang, Gang Combinatorics 05C10 Let $G$ be an embedded graph and $A$ an edge subset of $G$. The partial dual of $G$ with respect to $A$, denoted by $G^A$, can be viewed as the geometric dual $G^*$ of $G$ over $A$. If $A=E(G)$, then $G^A=G^*$. Denote by $γ(G^A)$ the genus of the embedded graph $G^A$. The maximum partial-dual genus of $G$ is defined as $$^\partialγ_{M}(G):=\max_{A \subseteq E(G)}γ(G^A).$$ For any planar graph $G$, it had been proved that $^\partialγ_{M}(G)$ does not rely on the embeddings of $G$. In this paper, we further prove that if $G$ is a connected planar graph of order $n\geq 2$, then $^{\partial}γ_{M}(G)\geq \frac{n-n_2-2n_1}{2}+1$, where $n_i$ is the number of vertices of degree $i$ in $G$. As a consequence, if $G$ is a connected planar graph of order $n$ with minimum degree at least 3, then $^{\partial}γ_{M}(G) \geq \frac{n}{2}+1$. Denote by $G^c$ the complement of a graph $G$ and by $χ(G^c)$ the chromatic number of $G^c$. Moreover, we prove that if $G \ncong K_4$ is a $λ$-edge-connected planar graph of order $n$, then $^{\partial}γ_{M}(G) \geq f(n,λ,χ(G^c))$, where $f(n,λ,χ(G^c))$ is a function of $n$, $λ$ and $χ(G^c)$. The first lower bound is tight for any $n$, and the second lower bound is tight for some 3-edge-connected graphs. |
| title | On the maximum partial-dual genus of a planar graph |
| topic | Combinatorics 05C10 |
| url | https://arxiv.org/abs/2503.20329 |