8 problems

Let $J_n(x)$ be the finite linear Jaco graph of order $n$, and let $\Delta(J_n(x))$ denote its maximum degree. A sequence is $p$-graphical for a graph family $\mathcal{F}$ when it…

Let $J_n(x)$ be a finite linear Jaco graph and let $\Delta(J_n(x))_{\mathrm{unique}}$ denote a maximum degree that occurs uniquely in the relevant sequence of maximum degrees. A se…

Let $J_\infty(x)$ be the infinite linear Jaco graph, and let $X$ be a $\gamma$-set, that is, a minimum dominating set. The vertex subscripts in the displayed set are…

Let $J_n(x)$ be a linear Jaco graph of order $n\geq1$. A diam-path is a path of diameter length, and let $P_d$ be a primary minimal dom-path, meaning a minimal path from $v_1$ to…

Let $J_n(x)$ be a finite linear Jaco graph and let $X$ be its $\Delta$-set. Non-singleton-Delta-vertex conjecture. The vertex subscripts $i$ for which $v_i$ does not yield a single…

Let $J_n(x)$ be a finite linear Jaco graph of order $n\geq3$, and let $X$ be its $\Delta$-set, equivalently its Jaconian set, consisting of vertices attaining the maximum degree. A…

Generalised Jaco graph conjecture. For every such Jaco graph,

Let $J_n(1)$ be a Jaco graph with $n\in\Bbb N$ and $n\geq 3$, and let $d^+(v_n)$ denote the out-degree of its vertex $v_n$. The McPherson number $\Upsilon(J_n(1))$ is said to be un…