Chromatic Gallai identities operating on Lovász number

Abstract

If \(G\) is a triangle-free graph, then two Gallai identities can be written as \(\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)\), where \(\alpha \) and \(\overline{\chi }\) 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 \(\Phi \) which associates to any graph parameter \(\beta \) such that \(\alpha (G) \le \beta (G) \le \overline{\chi }(G)\) for all graph \(G\), a graph parameter \(\Phi _\beta \) such that \(\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)\) for all graph \(G\). We prove that \(\vartheta (G) \le \Phi _\vartheta (G)\) and that \(\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)\) for all graph \(G\), where \(\vartheta \) is Lovasz theta function and \(\overline{\chi }_f\) is the fractional clique-partition number. Moreover, \(\overline{\chi }_f(G) \le \Phi _\vartheta (G)\) for triangle-free \(G\). Comparing to the previous strengthenings \(\Psi _\vartheta \) and \(\vartheta ^{+ \triangle }\) of \(\vartheta \), numerical experiments show that \(\Phi _\vartheta \) is a significant better lower bound for \(\overline{\chi }\) than \(\vartheta \).