5 problems

Let $G$) be a cubic graph. The Bickle–Phillips conjecture. … If $G$ does not contain $K_4$, then … If $G$ does not contain $K_4-e$, then … The first assertion is known, while the t…

Let $G$ be a planar graph with maximum degree $\Delta(G)$. Let $\tau_2(G)$ denote its 2-tone chromatic number. Cranston–LaFayette's conjecture. There exists a constant $C$ such tha…

Let $G$ be a cubic Halin graph of order $n$, where a Halin graph is formed from a tree with no vertices of degree two and a cycle joining its leaves in their planar cyclic order. T…

Let $G$ be a subcubic outerplanar graph, meaning that $G$ is outerplanar and has maximum degree at most $3$. For each positive integer $t$, let $\tau_t(G)$ denote the smallest numb…

For an integer $t\geq 2$, let $\tau_t(H)$ denote the $t$-tone chromatic number of a graph $H$, and let $C_n$ and $P_n$ denote the cycle and path on $n$ vertices, respectively. Cycl…