5 problems

Unfriendly partition conjecture. Every countable graph admits an unfriendly bipartition.

Countability-free vertex-flame conjecture. We may omit the countability of $D$ in Theorem.

Let $D=(V,A)$ be a digraph, let $\kappa$ be an infinite cardinal, and let $\mathcal{B}_i=(V_i,A_i)$ for $i<\kappa$ be edge-disjoint branchings in $D$. Define … Suppose that…

Let $G$ be a locally finite graph, and let $L(G)$ denote its line graph. A Hamiltonian circle is a circle in the Freudenthal compactification of $L(G)$ that contains every vertex o…

Bruhn's conjecture. Every $4$-connected locally finite planar graph has a Hamilton circle.