A sufficient condition for a subcubic graph having a 2-distance vertex 4-coloring

Question

Let $G$ be a subcubic graph with only vertices of degree 1 or degree 3.

Suppose that $G$ has an edge coloring $\varphi$ using colors from $\{1,2,3,4,5\}$ such that

• each edge is colored with a set of two elements from $\{1,2,3,4,5\}$ (e.g., $\varphi(e)=\{1,2\}$ for some edge $e$),
• if $v$ is a vertex of degree 3 that is incidenct with edges $e_1, e_2$ and $e_3$, then there exists $\{a,b,c\}\subseteq \{1,2,3,4,5\}$ such that $\varphi(e_1)=\{a,b\}$, $\varphi(e_1)=\{a,c\}$, $\varphi(e_1)=\{b,c\}$,
• if the distance between two edges $e_1$ and $e_2$ is 2 (i.e, $e_1$ and $e_2$ are not adjacent, and there is an edge $e_3$ adjacent to both $e_1$ and $e_2$), then $\varphi(e_1)\neq \varphi(e_2)$.

Then, does $G$ have a 2-distance vertex 4-coloring (i.e., a proper vertex 4-coloring of $G$ such that every two vertices at distance 2 receive distinct colors)?

Answer 1

The answer is false.

Let $G$ be the 3-prism and $\varphi$ the coloring shown below.

$G$ cannot have a 2-distance vertex 4-coloring. As all pairs of vertices in $G$ have distance either 1 or 2, a 2-distance coloring of $G$ would require 6 colors.