Spanning trees in sparse expanders

Given integers $n\ge \Delta\ge 2$, let $\mathcal{T}(n, \Delta)$ be the collection of all $n$-vertex trees with maximum degree at most $\Delta$. A question of Alon, Krivelevich and Sudakov in 2007 asks for determining the best possible spectral gap condition forcing an $(n, d,\lambda)$-graph to be $\mathcal{T}(n, \Delta)$-universal, namely, it contains all members of $\mathcal{T}(n, \Delta)$ as a subgraph simultaneously. In this paper we show that for all $\Delta\in \mathbb{N}$ and sufficiently large $n$, every $(n, d,\lambda)$-graph with $$\lambda\le\frac{d}{2\Delta^{5\sqrt{\log n}}}$$ is $\mathcal{T}(n, \Delta)$-universal. As an immediate corollary, this implies that Alon's ingenious construction of triangle-free sparse expander is $\mathcal{T}(n, \Delta)$-universal, which provides an explicit construction of such graphs and thus solves a question of Johannsen, Krivelevich and Samotij. Our main result is formulated under a much more general context, namely, the $(n,d)$-expanders. More precisely, we show that there exist absolute constants $C,c>0$ such that the following statement holds for sufficiently large integer $n$. $(1)$ For all $\Delta\in \mathbb{N}$, every $(n, \Delta^{5\sqrt{\log n}})$-expander is $\mathcal{T}(n, \Delta)$-universal. $(2)$ For all $\Delta\in \mathbb{N}$ with $\Delta \le c\sqrt{n}$, every $(n, C\Delta n^{1/2})$-expander is $\mathcal{T}(n, \Delta)$-universal. Both results significantly improve a result of Johannsen, Krivelevich and Samotij, and have further implications in locally sparse expanders and Maker-Breaker games that also improve previously known results drastically.Comment: 27 pages, 4 figures, comments are welcom

Clique immersion in graphs without fixed bipartite graph

A graph $G$ contains $H$ as an \emph{immersion} if there is an injective mapping $\phi: V(H)\rightarrow V(G)$ such that for each edge $uv\in E(H)$, there is a path $P_{uv}$ in $G$ joining vertices $\phi(u)$ and $\phi(v)$, and all the paths $P_{uv}$, $uv\in E(H)$, are pairwise edge-disjoint. An analogue of Hadwiger's conjecture for the clique immersions by Lescure and Meyniel states that every graph $G$ contains $K_{\chi(G)}$ as an immersion. We consider the average degree condition and prove that for any bipartite graph $H$, every $H$-free graph $G$ with average degree $d$ contains a clique immersion of order $(1-o(1))d$, implying that Lescure and Meyniel's conjecture holds asymptotically for graphs without fixed bipartite graph.Comment: 2 figure

Machine Learning-Enabled Regional Multi-Hazards Risk Assessment Considering Social Vulnerability

The regional multi-hazards risk assessment poses difficulties due to data access challenges, and the potential interactions between multi-hazards and social vulnerability. For better natural hazards risk perception and preparedness, it is important to study the nature-hazards risk distribution in different areas, specifically a major priority in the areas of high hazards level and social vulnerability. We propose a multi-hazards risk assessment method which considers social vulnerability into the analyzing and utilize machine learning-enabled models to solve this issue. The proposed methodology integrates three aspects as follows: (1) characterization and mapping of multi-hazards (Flooding, Wildfires, and Seismic) using five machine learning methods including Naïve Bayes (NB), K-Nearest Neighbors (KNN), Logistic Regression (LR), Random Forest (RF), and K-Means (KM); (2) evaluation of social vulnerability with a composite index tailored for the case-study area and using machine learning models for classification; (3) risk-based quantification of spatial interaction mechanisms between multi-hazards and social vulnerability. The results indicate that RF model performs best in both hazard-related and social vulnerability datasets. The most cities at multi-hazards risk account for 34.12% of total studied cities (covering 20.80% land). Additionally, high multi-hazards level and socially vulnerable cities account for 15.88% (covering 4.92% land). This study generates a multi-hazards risk map which show a wide variety of spatial patterns and a corresponding understanding of where regional high hazards potential and vulnerable areas are. It emphasizes an urgent need to implement information-based prioritization when natural hazards coming, and effective policy measures for reducing natural-hazards risks in future

On powers of Hamilton cycles in Ramsey-Tur\'{a}n Theory

We prove that for $r\in \mathbb{N}$ with $r\geq 2$ and $\mu>0$, there exist $\alpha>0$ and $n_{0}$ such that for every $n\geq n_{0}$, every $n$-vertex graph $G$ with $\delta(G)\geq \left(1-\frac{1}{r}+\mu\right)n$ and $\alpha(G)\leq \alpha n$ contains an $r$-th power of a Hamilton cycle. We also show that the minimum degree condition is asymptotically sharp for $r=2, 3$ and the $r=2$ case was recently conjectured by Staden and Treglown.Comment: 19 pages, 4 figure