On 4-critical t-perfect graphs

Abstract

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 $\Pi$ (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 $W_5$. 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 $P_6$-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 $P_6$-free graph is 4-colorable. Finally, we use results of Chudnovsky et al to show that $\overline{L(W_5)}$ and $\overline{L(\Pi)}$ are also the only 4-critical $t$-perfect $P_6$-free graphs.