3 problems

Let $G$ be a $d$-regular graph on $n$ vertices, let $H$ be a spanning subgraph of $G$, and suppose that $d=o(n)$. For each $k\in\{0,1,\dots,d\}$, write $\#\{v\in V(H):d_H(v)=k\}$ f…

Let $k$ be a positive integer, and let $T$ be a strongly $k$-connected tournament on $n$ vertices. A strongly $k$-connected spanning subgraph is a spanning subgraph with strong ver…

Let $G=(V,E)$ be a finite graph, and let $f,g:V\rightarrow\mathbb{N}$ be any two functions. Fix the vertex activities $\mathbf{u}$ as in $(4.1)$. Let $N_j$ denote the coefficients…