On Chromatic no. of 3K1-free graphs and R(3, k)

TL;DR

This paper establishes bounds on the chromatic number of 3K1-free graphs with small independence number and conjectures general formulas relating to Ramsey numbers R(3, omega).

Contribution

It provides new upper bounds for the chromatic number of certain 3K1-free graphs and proposes conjectures linking these bounds to Ramsey numbers R(3, omega).

Findings

Proved bounds for chromatic number when independence number is 2 and omega ≤ 11.

Formulated conjectures for general omega relating chromatic number and R(3, omega).

Verified conjectures for omega ≤ 9.

Abstract

Here we prove that if G has independence no. 2 and clique size omega with omega less than or equal to 11, then (1) chromatic no. is less than or equal to (omega2+12omega-13)/8, if omega is odd, and (2) chromatic no. is less than or equal to (omega2+10omega)/8, if omega is even. We further conjecture that the results are true in general for all omega. We also conjecture that (A) if omega is odd and R(3, omega) is even, then R(3, omega) = (omega2+8omega-9)/4, (B) if omega and R(3, omega) are both odd, then (omega2+8omega-13)/4, (C) if omega and R(3, omega) are both even, then R(3, omega) = (omega2+6omega)/4 and (D) if omega is even and R(3, omega) is odd, then (omega2+6omega-4)/4. Again we verify the results for omega less than or equal to 9.