We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the Gauss-Bonnet theorem involving boundary turns, we show that a graph is CP parabolic if \[ \sum_{n=1}^\infty \frac{1}{\sum_{j=0}^{n-1} (k_j +6)} = \infty, \] where $k_n$ degree excess sequence defined...

Simple drawings are of graphs in the plane or on sphere such that vertices distinct points, edges Jordan arcs connecting their endpoints, and intersect at most once (either a proper crossing shared endpoint). generalized twisted if there is point O every ray emanating from crosses edge drawing which exactly once. We show all $$K_n$$ contain $$2n-4$$ empty triangles, by this making substantial s...

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of complete graph $K_n$ in plane (equivalently, sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved expected crossings a spherical drawing which points are randomly distributed and joined by geodesics precisely $n^4/64+O(n...

Given a partial edge coloring of complete graph $K_n$ and lists allowed colors for the non-colored edges $K_n$, can we extend to proper using only from lists? We prove that this question has positive answer in case when both color satisfy certain sparsity conditions.

The notion of weak saturation was introduced by Bollob\'as in 1968. Let $F$ and $H$ be graphs. A spanning subgraph $G \subseteq F$ is weakly $(F,H)$-saturated if it contains no copy but there exists an ordering $e_1,\ldots,e_t$ $E(F)\setminus E(G)$ such that for each $i \in [t]$, the graph \cup \{e_1,\ldots,e_i\}$ a $H'$ $e_i H'$. Define $wsat(F,H)$ to minimum number edges graph. In this paper,...

Consider a sequence of positive integers $$\{k_n,n\ge 1\}$$ , and an array nonnegative real numbers $$\{a_{n,i},1\le i\le k_n,n\ge satisfying $$\sup _{n\ge 1}\sum _{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty ).$$ This paper introduces the concept $$\{a_{n,i}\}$$ -stochastic domination. We develop some techniques concerning this apply them to remove assumption in strong law large Chandra Ghosal (Acta Ma...

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ odd prime number. $K_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $K_n$ denote its $n$-th layer. The Mordell--Weil is said to constant in tower if for all $n$, $E(K_n)$ equal $E(K)$. We apply techniques Iwasawa theory obtain explicit conditions above sense. then indicate potential applications Hilbert's ten...