On the maximum partial-dual genus of a planar graph
Jiaying Chen, Xian'an Jin, Gang Zhang
TL;DR
This work studies the maximum partial-dual genus $^ abla\partial\gamma_{M}(G)$ of connected planar graphs, showing it is embedding-independent and deriving new lower bounds in terms of the order and degree sequence, e.g., $^ abla\partial\gamma_{M}(G)\ge \frac{n-n_2-2n_1}{2}+1$. It extends the toolkit for partial duals by employing the $y_G$ parameter, Xuong-type decompositions, and the $G_A$ construction, to obtain bounds that also incorporate edge-connectivity and the chromatic number of the complement, with tightness results for various regimes. A central contribution is the bound $^ abla\partial\gamma_{M}(G)\ge f(n,\lambda,\chi(G^c))$ for $\lambda$-edge-connected planar graphs with $G\ncong K_4$, expanding the known landscape of extremal partial-dual genus results. The findings illuminate how planarity, local degree structure, and complement colorability constrain partial-dual genus and suggest structural characterizations and conjectures for tightness across graph families.
Abstract
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.