On 4-critical t-perfect graphs

It is an open question whether the chromatic number of t-perfect graphs is bounded by a constant. The largest known value for this parameter is 4, and the only example of a 4-critical t-perfect graph, due to Laurent and Seymour, is the complement of the line graph of the prism Π (a graph is 4-critical if it has chromatic number 4 and all its proper induced subgraphs are 3-colorable). In this paper, we show a new example of a 4-critical t-perfect graph: the complement of the line graph of the 5-wheel W5. Furthermore, we prove that these two examples are in fact the only 4-critical t-perfect graphs in the class of complements of line graphs. As a byproduct, an analogous and more general result is obtained for h-perfect graphs in this class. The class of P6-free graphs is a proper superclass of complements of line graphs and appears as a natural candidate to further investigate the chromatic number of t-perfect graphs. We observe that a result of Randerath, Schiermeyer and Tewes implies that every t-perfect P6-free graph is 4-colorable. Finally, we use results of Chudnovsky et al to show that L(W5) and L(Π) are also the only 4-critical t-perfect P6-free graphs.