The sandwich line graph

We observe that ω ( G ) + χ ( S ( G → ) ) = n = ω ( S ( G → ) ) + χ ( G ) for any graph G with n vertices, where G → is any acyclic orientation of G and where S ( G → ) is the (complement of the) auxiliary line graph introduced in [Cornaz, D., and Jost, V., A one-to-one correspondence between colorings and stable sets, Operations Research Letters 36 (2008), 673–676]. (Where as usual, ω and χ denote the clique number and the chromatic number.) It follows that, for any graph parameter β ( G ) sandwiched between ω ( G ) and χ ( G ) , then Φ β ( G → ) : = n − β ( S ( G → ) ) is sandwiched between ω ( G ) and χ ( G ) too. Numerical experiments show that Φ ϑ is closer to χ than ϑ, where ϑ is Lovász theta function.