7 problems

Subgraph core conjecture. The subgraph core of $G$ is contained in a block of $G$.

Let $\mathcal{B}(n,q)$ be the class of block graphs with parameters $n$ and $q$, let $P_b^q$ denote the corresponding $q$-clique path with $b$ blocks, and let $\prec$ be the graph…

Let $G$ be a block graph. Write $\omega(G)$ for its clique number, $\alpha_{\min}(G)$ for the minimum size of a maximal independent set, and $\chi_{=}(G)$ for its equitable chromat…

Let $n\geq 5$, and let $G$ be a block graph of order $n$. Suppose that $G$ has maximum mean CIS order among all block graphs of order $n$. Block-graph caterpillar conjecture. Then…

Nonsingular edge-extension conjecture. If an edge is added between vertices $v_1\rtimes V(G_1)$ and $v_2\rtimes V(G_2)$, then the resulting block graph is nonsingular.

Nullity conjecture. If each block of $G$ has order at least $3$, then the nullity of $G$ is at most $1$.

Let $\mathsf{C}_n$ be a uniform random graph from a block class, with each block receiving weight $0$ or $1$, and let $t_n$ be a sequence tending to infinity. McDiarmid and Scott's…