Talk:Line perfect graph

Characterization of line perfect graphs is incorrect?

In the first paragraph of this article, it says that line perfect graphs "are the graphs in which every odd-length simple cycle is a triangle". I don't think this is generally true. Here's why. Take the graph G composed of the disjoint union of a 5-cycle and a 3-cycle, whose line graph L(G) is itself (because cycles are their own line graphs). The chromatic number of L(G) is equal to the clique number of L(G), three (color both components of L(G) with three colors each), so L(G) is perfect, so G is line perfect, even though G has an odd-length simple cycle that is not a triangle (the cycle of length 5). Adam El-Sawaf (talk) 23:15, 18 November 2024 (UTC)

Being perfect requires that chromatic number = clique number on all induced subgraphs, not just on the graph itself. For the induced subgraph consisting of the 5-cycle but not of the 3-cycle, this condition is not met. —David Eppstein (talk) 23:40, 18 November 2024 (UTC)

Ah shoot, you're right, I forgot about that part of the condition. That's just for weak perfection, yeah. Thanks! Adam El-Sawaf (talk) 23:44, 18 November 2024 (UTC)