7 problems

Let $D$ be a digraph with maximum out-degree $\Delta^+(D)$, biclique number $\overset{\leftrightarrow}{\omega}(D)$, and dichromatic number $\vec{\chi}(D)$. Write…

Let $D$ be a digraph, let … let $\overset{\leftrightarrow}{\omega}(D)$ be the biclique number, and let $\vec{\chi}(D)$ be the dichromatic number. Write…

Let $D$ be a digraph. Define its maximum geometric-mean degree by … let $\overset{\leftrightarrow}{\omega}(D)$ be its biclique number, and let $\vec{\chi}(D)$ be its dichromatic nu…

Let $D$ be a digraph on $n$ vertices, let $\mathcal{D}_k(D)$ be the graph whose vertices are the $k$-dicolourings of $D$, and let $\delta^*_{\min}(D)$ denote the min-degeneracy of…

Let $C_3$ denote the directed 3-cycle, let $S^+_2$ denote the oriented forest appearing in the source, and let $\overleftrightarrow{K_2}$ be the digraph consisting of two opposite…

Let $H$ be a hero, meaning a tournament such that every tournament not containing $H$ has bounded dichromatic number, and let $F$ be an oriented forest. A set of digraphs is heroic…

Let $D$ be an oriented planar graph, meaning a planar digraph without directed cycles of length at most $2$. A $2$-colouring of $D$ is a function $f:V(D)\to\{1,2\}$ such that the s…