24 problems

Let $G$ be a graph with independence number $\alpha(G)=2$, let $\chi(G)$ denote its chromatic number, and let $K_{\chi(G)}$ be the complete graph on $\chi(G)$ vertices. A weak imme…

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $K_{\chi(G)}$ be the complete graph on $\chi(G)$ vertices. A strong immersion of a graph $H$ in $G$ consists of…

Weak knitwork immersion conjecture. The class $\mathbf{T}(k,d,\Delta,\ell;\Omega)$ is well-quasi-ordered by $\Omega$-knitwork immersion.

Weak immersion conjecture. The class of Eulerian digraphs is well-quasi-ordered by weak immersion.

Johnson's conjecture. For every $d\geq 1$, the class of Eulerian digraphs of maximum degree $2d$ is well-quasi-ordered by strong immersion.

Let $t\in\mathbb{Z}^+$ and let $G$ be a graph. An immersion of $K_t$ is strong odd if its paths are pairwise edge-disjoint, have odd length, and no terminal is an interior vertex o…

Let $G$ be a graph and let $G\xrightarrow{\operatorname{odd}} K_{ct}$ mean that $G$ admits a $t'$-oddomorphism to $K_{ct}$, as defined in the source. Linear immersion conjecture. T…

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be two distinct immersion-closed and union-closed graph classes. For graphs $G$ and $H$, write $G =_{\mathcal{M}} H$ when…

Let $G$ be a finite, undirected, loopless graph, and let $\chi(G)$ denote its chromatic number. A graph $H$ is a totally odd immersion of $G$ when the edges of $H$ are represented…

For a graph $G$, let $\operatorname{toi}(G)$ be the maximum integer $t$ such that $G$ contains a totally odd strong immersion of $K_t$, and let $\chi(G)$ denote its chromatic numbe…

Let $G$ be a graph, let $\chi(G)$ denote its chromatic number, and let $K_t$ be the complete graph on $t$ vertices. An $H$-immersion is an injective mapping of $V(H)$ to $V(G)$ tog…

Let $G$ be a graph, let $\chi(G)$ be its chromatic number, and let $K_k$ denote the complete graph on $k$ vertices. Clique-immersion conjecture. If … then $G$ contains a $K_k$-imme…

Totally odd strong immersion conjecture. Every graph $G$ with $\chi(G)\ge t$ contains a totally odd strong immersion of $K_t$.

Churchley's conjecture. Every graph $G$ with $\chi(G)\ge t$ contains a totally odd immersion of $K_t$.

Let $D$ be a digraph, let $k$ be a positive integer, and write $\delta^{+}(D)$ for its minimum out-degree. A transitive tournament on $k$ vertices is the tournament whose vertices…

For integers $m\geq 3$, let $K_{m+1}$ be the complete graph on $m+1$ vertices, let $\mu_m(K_{m+1})$ be its $m$-Mycielskian, and let $\operatorname{im}(G)$ denote the largest $t$ su…

For a graph $H$, let $\chi_*(H)$ denote the strong-immersion coloring parameter used in the paper, and let the clustered chromatic number of a graph class be the least number of co…

Let $t$ be a positive integer. For a graph $H$, an $H$-immersion is a graph isomorphic to a graph obtained from a subgraph by repeatedly splitting off pairs of edges with a common…

Let $G$ and $H$ be graphs, and write $\operatorname{im}(G)=t$ and $\operatorname{im}(H)=r$. Strong-product immersion conjecture. Then … This conjecture is motivated by the identity…

Let $G$ and $H$ be graphs, and write $\operatorname{im}(G)=t$ and $\operatorname{im}(H)=r$. Direct-product immersion conjecture. Then … This conjecture asserts that the direct prod…

Let $G$ be an $n$-vertex graph, and let $\alpha(G)$ denote its independence number. Independence-number clique immersion conjecture. The graph $G$ contains … as an immersion. This…

Let $t$ be a positive integer, and let $G$ be a graph. An immersion of $K_t$ in $G$ consists of distinct branch vertices corresponding to the vertices of $K_t$, together with paths…

Let $G$ be a graph on $n$ vertices, and let $\alpha(G)$ denote its independence number. An immersion of a graph $H$ in $G$ is a subgraph obtainable from $G$ by lifting edges and de…

Immersion conjecture for transitive tournaments. There exists a function $f$ so that every simple digraph of minimum outdegree $f(t)$ contains an immersion of the transitive tourna…