arXiv subjects

Miaoxia Zhuang

Publications and source records attributed to Miaoxia Zhuang.

4 recordsLinked to original sources

arXiv2026

On minimal nonperfectly divisible fork-free graphs

A fork is a graph obtained from $K_{1,3}$ (usually called claw) by subdividing an edge once. A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. In this paper, we prove that the perfect divisibility of fork-free graphs is equivalent to that of claw-free graphs. We also prove that, for $F\in \{P_7, P_6\cup K_1\}$, each (fork, $F$)-free graph $G$ is perfectly divisible and hence $χ(G)\leq \binom{ω(G)+1}{2}$.

math.CO↗

arXiv2026

Some properties of minimally nonperfectly divisible graphs

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$, and a graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. A clique $X$ of a connected graph $G$ is called a clique cutset if $G-X$ is disconnected. In this paper, we investigate the relationship between the perfect divisibility of a graph and its perfect weighted divisibility. We also show that $2P_3$-free or claw-free minimally nonperfectly divisible graphs contain no clique cutset, that conditionally answers a question of Hoàng [Discrete Math. \textbf{349} (2025) 114809].

math.CO↗

arXiv2026

Perfect weighted divisibility is equivalent to perfect divisibility

A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. A graph $G$ is perfectly weight divisible if for every positive integral weight function on $V(G)$ and each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. In this paper, we prove that the perfect divisibility of a graph is equivalent to its perfect weighted divisibility.

math.CO↗

arXiv2025

Perfect divisibility of (fork, antifork$\cup K_1$)-free graphs

A {\em fork} is a graph obtained from $K_{1,3}$ (usually called {\em claw}) by subdividing an edge once, an {\em antifork} is the complement graph of a fork, and a {\em co-cricket} is a union of $K_1$ and $K_4-e$. A graph is perfectly divisible if for each of its induced subgraph $H$, $V (H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. Karthick {\em et al.} [Electron. J. Comb. 28 (2021), P2.20.] conjectured that fork-free graphs are perfectly divisible, and they proved that each (fork, co-cricket)-free graph is either claw-free or perfectly divisible. In this paper, we show that every (fork, {\em antifork}$\cup K_1$)-free graph is perfectly divisible. This improves some results of Karthick {\em et al.}.

math.CO↗