5 problems

For a graph $G$, write $chi^*(G)$ for its fractional chromatic number and $\operatorname{oddgirth}(G)$ for the length of its shortest odd cycle; a t-minor is the graph-minor operat…

A graph is t-perfect when its stable set polytope is described by nonnegativity, clique, and odd-cycle inequalities. A graph is triangle-free when it has no three mutually adjacent…

Four-colourability conjecture. Every t-perfect graph is four-colourable.

Let $G$ be a $t$-perfect graph, meaning that its stable set polytope is determined by non-negativity, edge, and odd-cycle inequalities. Four-colour conjecture for $t$-perfect graph…

A graph is $t$-perfect if it satisfies the defining $t$-perfection property for graphs. Shepherd–Sebő's four-colour conjecture. Every $t$-perfect graph is $4$-colourable. This is a…