14 problems

Mixed-graph threshold orientation conjecture. There is a constant $k>2$ such that every mixed graph with minimum pseudo dicut size at least $k$ has an orientation $E^+$ satisfying

Maximum-order conjecture. All these mixed graphs have the maximum possible order among mixed graphs with the same specified parameters.

Let $(m,n)$ be nonnegative integers with $(m,n)\neq(0,1)$. An $(m,n)$-mixed graph is a graph whose edges may carry $m$ directed and $n$ undirected edge types, and a simple $(m,n)$-…

Let $H=(V,E,A)$ be a mixed graph, let $\mathcal{D}(V)$ denote the relevant family of subpartitions, and let $d_A^-(X)$ count arcs entering $X$. Let $k\geq0$ and $d\geq1$ be integer…

Let $CMRM(m,n,c)$ be a chordal $m$-ring mixed graph on vertex set $\mathbb Z_m\times\mathbb Z_n$, with the arcs and chords defined by the source, and let $k$ be an even diameter sa…

Let $CRM(N,c)$ be a chordal ring mixed graph with vertex set $\mathbb Z_N$, directed-cycle arcs $i\mathbin{\rightarrow}i+1$, and chords determined by the odd parameter $c$ as in th…

Let $F=(V,E\cup A)$ be a mixed graph, let $k\in\mathbb Z_{>0}$, and let $\ell:K\rightarrow\mathbb Z_{>0}$. For a subpartition $\mathcal{P}$ of $V$, write…

Total-regularity conjecture. Any mixed graph with excess one is totally regular.

Let $G\in\mathcal{G}_n^{sc}$ be a self-converse mixed graph, let $W=W(G)$ be its walk-matrix, let $\mathscr{U}_G$ be the associated set of Gaussian rational unitary matrices, and l…

Let $G$ be a self-converse mixed graph of order $n$, with walk-matrix … Here $\mathcal{G}_n^{sc}$ denotes the self-converse mixed graphs of order $n$, and $G$ is DGS when it is det…

Total regularity conjecture. All $(r,z,k;-1)$- and $(r,z,k;+1)$-graphs are totally regular.

Bosák's conjecture. A finite graph $G$ is a mixed Moore graph of degree $d$ and diameter $k$ if and only if either $d=1$ and $G$ is $\vec{C}_{k+1}$, or $d=2$ and $G$ is $C_{k+1}$.

Let $K_{m,n}$ be the complete bipartite graph with positive integers $m$ and $n$. A mixed graph is DHS if it is determined by its Hermitian adjacency spectrum among mixed graphs. F…

Conjecture on maximal SGs. Maximal SGs possess the same statistical properties that both maximal ancestral graphs and SGs possess.