The corona–kernel bound for graphs with odd cycles

Let $G$ be a graph. Write $\textnormal{corona}(G)$ for its corona, \textnormal{\ker}(G) for its kernel, and let $\alpha(G)$ denote its independence number. Let $k$ be the number of odd cycles in $G$.

Corona–kernel conjecture. For every graph $G$,

\left|\textnormal{corona}(G)\right|+\left|\textnormal{\ker}(G)\right|\le 2\alpha(G)+k.

This conjecture extends the corresponding equality and inequality phenomena for corona and core in König–Egerváry graphs and general graphs. Its resolution is not indicated in the supplied text.