3 problems

Let $k\geq 3$ be an integer, let $G$ be a graph, and for a vertex set $X\subseteq V(G)$ let $c_{2j+1}(G-X)$ denote the number of components of $G-X$ having order $2j+1$. A {…

Let $k\geq 3$, and let $G$ be a graph. For a graph $H$, let $\mathcal{C}(H)$ be the set of components of $H$; for $i\geq 1$, let…

Let $G$ be a simple $(3,4)$-biregular bigraph, meaning a bipartite graph whose vertices in one part have degree $3$ and whose vertices in the other part have degree $4$. A proper p…