7 problems

Weakly supercritical circumference conjecture.

Linear circumference conjecture.

Genus-one twist-operator conjecture. In genus $1$, there might be a way to construct a model in which the twist operator is present, unlike the straightforward generalisation to hi…

Let $P(n,M)$ be the random planar graph with $n$ vertices and $M$ edges, let $\chi(P(n,M))$ denote its chromatic number, and let $K_4$ be the complete graph on four vertices. Chrom…

Let $k$ and $ell$ be the parameters used for the random planar graph model, with $\ell\leq k$. Let Theorem(iv) denote the assertion concerning the average total size of the trees r…

Let $n$ denote the number of components in a spanning forest, and let $\epsilon$ measure the distance from the critical coupling through $g=g_c e^{-\epsilon}$. For the cases analys…

Consider the spanning-forest model on random planar graphs, with coupling parameter $t$ and graph degree fixed arbitrarily. The preceding construction uses the graph's cyclic edge…