4 problems

Goddard–Henning conjecture. If

Let $d\geq 2$, and let $G$ be a connected bipartite graph of order $n$. Write $C_{2d+2}$ for the cycle of length $2d+2$, and let $\mathcal{B}_d$ denote the graph family defined in…

Verstraëte's conjecture. If $G$ is a cubic graph with girth at least $6$, then

Let $G$ and $H$ be graphs, and let $i(G)$ denote the smallest cardinality of an independent dominating set in $G$, equivalently the cardinality of a smallest maximal independent se…