arXiv subjects
Publications and source records attributed to Yuwen Zhou.
The well-known Erdős-Hajnal conjecture states that for any graph $H$, there is a constant $c=c(H)>0$ such that every $n$-vertex graph $G$ with no induced copies of $H$ contains a clique or an independent set of size at least $n^{c}$. In this paper, we prove that the Erdős-Hajnal conjecture holds for every path.
In this paper we prove that every graph $G$ with no induced six-vertex path satisfies $χ(G)\le ω(G)^{c\log_2ω(G)}$ for $c=45/\log_2(4/3)<109$. The best previous bound was exponential in $ω(G)$. Our result is the first quasi-polynomial bound for $P_6$-free graphs and generalizes the bound for $P_5$-free graphs by Scott, Seymour, and Spirkl [Polynomial bounds for chromatic number. {IV}. {A} near-polynomial bound for excluding the five-vertex path. {\em Combinatorica}, 43(5):845--852, 2023].
Recently bosonic quantum Hall droplets have been observed in rapidly rotating two-dimensional Bose-Einstein condensates (BECs), which exhibit robust dynamical stability. Inspired by this, we systematically investigate the collision and coalescence dynamics of these droplets within the Gross-Pitaevskii framework. For two-droplet collisions, we find two distinct collision outcomes, namely merging and separation, that are controlled by the initial relative velocity. The critical velocity exhibits a universal scaling law with the interaction and the particle number as $v_c \propto (gN)^{1/4}$, which can be interpreted from a simplified analytical model, revealing the essential role of the collision time. It differs fundamentally from the mechanism governing the conventional Lee-Huang-Yang stabilized quantum droplets. Furthermore, while the collision can change the shape of the droplet significantly, the center of mass trajectory remains nearly unaffected, owing to the conservation of angular momentum. For overlapping stationary droplets, vortex arrays can emerge through Kelvin-Helmholtz instability driven by phase-induced shear flow. Although two droplets may merge into a larger one, extended states cannot be constructed from multiple overlapping droplets. Instead, the system dynamically reorganizes into new isolated droplets, revealing the localized property in the bulk region. Our results reveal the unique nonequilibrium dynamics of quantum Hall droplets and suggest new pathways for manipulating strongly correlated rotating quantum fluids.