13 problems

Unfriendly Partition Conjecture.

Let $G$ be a connected, infinite, locally finite $L_1$-graph on at least three vertices. Write $N(u)$ for the neighborhood of a vertex $u$, $d(u,v)$ for the graph distance, and let…

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

Let $G$ be an infinite graph. A homomorphic image of a graph $H$ in $G$ is the image of a graph homomorphism from $H$ to $G$. Taylor–Erdős conjecture. If $G$ has uncountable chroma…

Haslegrave's conjecture. Every graph, including uncountable graphs, is majority $3$-choosable.

Halin's end-degree conjecture. If $\deg(\varepsilon)=\kappa$, then $\varepsilon$ contains a subdivision of a graph of the form $S\times\omega$, where $S$ is a $\kappa$-star. Equiva…

Let $G$ be a complete bipartite graph whose two bipartition classes have the same infinite cardinality, and let the edges of $G$ be coloured with $r$ colours. A generalised path is…

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…

For a partial order type $\psi$ and a cardinal $\kappa$, write $P\nrightarrow(\psi)^1_\kappa$ when some coloring $c:P\to\kappa$ has no monochromatic subset of order type $\psi$; th…

An interval graph is the intersection graph of a family of non-empty convex subsets of a linear order. For a class of graphs, let $\mathcal{C}_{\chi}$ mean that, for every graph…

A poset is a set equipped with a partial order, and a chain is a subset whose elements are pairwise comparable. For a graph $G$, let $\theta(G)$ be the least cardinality of a cover…

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