Strong Ramsey saturation for cycles

22 May 2012
Jozef Skokan
We call a graph $H$ \emph{Ramsey-unsaturated} if there is an edge in the complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not change upon adding it to $H$. This notion was introduced by Balister, Lehel and Schelp who also showed that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add {\em any} chord without changing the Ramsey number of the cycle $C_n$, unless $n$ is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph $H$ is obtained by adding a linear number of chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum degree of $H$ is bounded, $H$ is either bipartite (for even $n$) or almost bipartite (for odd $n$), and $n$ is large. This motivates us to call cycles \emph{strongly} Ramsey-unsaturated. Our proof uses the regularity method.
  • Combinatorial Theory Seminar