Chromatic Gallai identities operating on Lovász number

If G is a triangle-free graph, then two Gallai identities can be written as α(G)+ χ(L(G)) = |V (G)| = α(L(G))+ χ(G), where α and χ denote the stability number and the clique-partition number, and L(G) is the line graph of G. We show that, surprisingly, both equalities can be preserved for any graph G by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of G. As a consequence, one obtains an operator which associates to any graph parameter β such that α(G) ≤ β(G) ≤ χ(G) for all graph G, a graph parameter β such that α(G) ≤ β(G) ≤ χ(G) for all graph G. We prove that θ(G) ≤ θ(G) and that χ f (G) ≤ χ f (G) for all graph G, where θ is Lovász theta function and χ f is the fractional clique-partition number. Moreover, χ f (G) ≤ θ(G) for triangle-free G. Comparing to the previous strengthenings θ and θ+ of θ , numerical experiments show that θ is a significant better lower bound for χ than θ .