Abstract
Goodman proved that the sum of the number of triangles in a graph on nodes and its complement is at least ; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdős conjectured that a similar inequality will hold for in place of , but this was disproved by Thomason. But an analogous statement does hold for some other graphs, which are called common graphs. Characterization of common graphs seems, however, out of reach. Franek and Rödl proved that is common in a weaker, local sense. Using the language of graph limits, we study two versions of locally common graphs. We sharpen a result of Jagger, Štovíček and Thomason by showing that no graph containing can be locally common, but prove that all such graphs are weakly locally common. We also show that not all connected graphs are weakly locally common.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
