7 problems

Let $\mathcal{C}$ be a $1$-colorful minor-closed class, meaning a class of $1$-colorful graphs closed under taking colorful minors. A $1$-colorful rainbow $(k\times k)$-grid is the…

Let $(G,\chi)$ be a $1$-colorful graph, and let … be the set of vertices with color $1$. A class of colorful graphs is colorful minor-closed if it is closed under taking colorful m…

Pure half-integral indegree conjecture. Every such spanning vertex $x$ has indegree $1$ at every Steiner node. Consequently, the PHI procedure is exhaustive for every pure half-int…

Ten-vertex gap conjecture. With $n\leq 10$, the highest integrality gap is

Six-vertex unit-gap conjecture. For $n=6$, the CM and BCR formulations have integrality gap equal to $1$.

PHI indegree conjecture. Every Steiner node of every PHI spanning vertex has indegree exactly one.

Let $G=(V,E)$ be the graph of the Steiner traveling salesman problem, with depot $1$, required-node set $V_R$, and $n_R=|V_R|$. For each $S\subseteq V\setminus\{1\}$ with…