The 3-colorability conjecture for triangle- and ISK4-free graphs

Here graphs are simple and finite. A graph is triangle-free if it contains no triangle and ISK4-free if it contains no induced subdivision of $K_4$, where $K_4$ is the complete graph on four vertices. A graph is 3-colorable if its vertices can be colored with three colors so that adjacent vertices receive different colors. The 3-colorability conjecture for triangle- and ISK4-free graphs. Every triangle-free, ISK4-free graph is 3-colorable. The paper states that this conjecture follows from the preceding low-degree vertex conjecture and proves both conjectures for graphs of girth at least $5$; the unrestricted case remains open in the supplied text.