Résumé en anglais

We observe that View the MathML source for any graph G with n vertices, where View the MathML source is any acyclic orientation of G and where View the MathML source 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 View the MathML source is sandwiched between ω(G) and χ(G) too. Numerical experiments show that Φtheta is closer to χ than theta, where theta is Lovász theta function