6 problems

For positive integers $\ell$ and $n$, let $p_\ell(n)$ be the maximum number of edges in an $n$-vertex graph containing no two vertices of equal degree connected by a path of length…

Let $G$ be a connected graph and let $u,v\tnobreak\text{ be vertices with }d(u),d(v)geq 2$. Suppose that $u$ and $v$ are joined by a path … where $d(w_i)=2$ for $1\leq i\leq s-1$.…

For a finite simple graph $G$, let $N(G,H)$ be the number of unlabeled, not necessarily induced copies of $H$ in $G$. Define … Let $P_k$ denote the path on $k$ vertices, so…

For positive integers $\ell$ and $n$, let $p_\ell(n)$ denote the maximum number of edges in an $n$-vertex graph containing no two vertices of equal degree connected by a path of le…

Odd-length extremal conjecture. For any odd integer $\ell\geq 3$ and sufficiently large $n$, it holds that

Let $F$ be an $(r+1)$-chromatic graph and let $T$ be a graph that does not contain $F$ as a subgraph. For a graph $H$, let $u(T,H)$ denote the number of distinct, not necessarily…