A coloring algorithm for 4K14K_1-free line graphs

When $F$ is a set of four-vertex graphs the complexity of the vertex coloring problem in the class of $F$-free graphs is known, with three exceptions: $F = \{\text{claw}, 4K_1\}$, $F = \{\text{claw}, 4K_1, \text{co-diamond}\}$, and $F = \{C_4, 4K_1\}$. We study the coloring problem for $\{\text{claw},4K_1\}$-free graphs. We solve the vertex coloring problem for a subclass of this class which contains the class of $4K_1$-free line graphs. Our result implies that the chromatic index of a graph with no matching of size four can be computed in polynomial time. Joint work with Dallas J. Fraser, Angele M. Hamel, Chinh T. Hoang (Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario, Canada)Non UBCUnreviewedAuthor affiliation: Laboratoire G-SCOPFacult