HMIC 2022 Problem 4

Call a simple graph $G$ quasi-colorable if we can color each edge blue, red, green, and white such that

• for each vertex $v$ of degree 3 in $G$, the three edges containing $v$ as an endpoint are either colored blue, red, and green, or all three edges are white,
• not all edges are white.

A connected graph $G$ has $a$ vertices of degree $4, b$ vertices of degree 3, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: "If $a / b>c$, then $G$ is quasi-colorable."