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Reed's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that everygraph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The secondauthor formulated a {\em local strengthening} of this conjecture that considersa bound supplied by the neighbourhood of a single vertex. Following the ideathat the chromatic number cannot be greatly affected by any particular stableset of vertices, we propose a further strengthening that considers a boundsupplied by the neighbourhoods of two adjacent vertices. We provide somefundamental evidence in support, namely that the stronger bound holds in thefractional relaxation and holds for both quasi-line graphs and graphs withstability number two. We also conjecture that in the fractional version, we canpush the locality even further.

A superlocal version of Reed's Conjecture