Planar Tur\'an Number of the Θ6\Theta_6

Let $F$ be a graph. The planar Tur\'an number of $F$, denoted by $\text{ex}_{\mathcal{P}}(n,F)$, is the maximum number of edges in an $n$-vertex planar graph containing no copy of $F$ as a subgraph. Let $\Theta_k$ denote the family of Theta graphs on $k\geq 4$ vertices, that is, a graph obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Y. Lan, et.al. determined sharp upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_4)$ and $\text{ex}_{\mathcal{P}}(n,\Theta_5)$. Moreover, they obtained an upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_6)$. They proved that, $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$. In this paper, we improve their result by giving a bound which is sharp. In particular, we prove that $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{48}{7}$ and demonstrate that there are infinitely many $n$ for which there exists a $\Theta_6$-free planar graph $G$ on $n$ vertices, which attains the bound.Comment: 23 pages, 19 figure

DP-MPM: Domain partitioning material point method for evolving multi-body thermal–mechanical contacts during dynamic fracture and fragmentation

We propose a material point method (MPM) to model the evolving multi-body contacts due to crack growth and fragmentation of thermo-elastic bodies. By representing particle interface with an implicit function, we adopt the gradient partition techniques introduced by Homel and Herbold (2017) to identify the separation between a pair of distinct material surfaces. This treatment allows us to replicate the frictional heating of the evolving interfaces and predict the energy dissipation more precisely in the fragmentation process. By storing the temperature at material points, the resultant MPM model captures the thermal advection–diffusion in a Lagrangian frame during the fragmentation, which in return affects the structural heating and dissipation across the frictional interfaces. The resultant model is capable of replicating the crack growth and fragmentation without requiring dynamic adaptation of data structures or insertion of interface elements. A staggered algorithm is adopted to integrate the displacement and temperature sequentially. Numerical experiments are employed to validate the diffusion between the thermal contact, the multi-body contact interactions and demonstrate how these thermo-mechanical processes affect the path-dependent behaviors of the multi-body systems

Book free 33-Uniform Hypergraphs

A $k$-book in a hypergraph consists of $k$ Berge triangles sharing a common edge. In this paper we prove that the number of the hyperedges in a $k$-book-free 3-uniform hypergraph on $n$ vertices is at most $\frac{n^2}{8}(1+o(1))$