The conjectured formula for maximal chordal subgraphs

Let $f(n,m)$ be the largest integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. For $k\geq 2$, let $t_k(n)$ denote the number of edges in the balanced complete $k$-partite graph on $n$ vertices. Let $n\geq 1$ and suppose that

$$t_k(n)+1\leq m\leq t_{k+1}(n).$$

The maximal chordal subgraph conjecture. Then

$$f(n,m)=\min_{t,r}\bigl(kn-t+r\bigr)-\binom{k+1}{2},$$

where the minimum is taken over all $t,r\geq 0$ satisfying

$$t_{k-1}(n-t)+t(n-t)+t_2(r)\geq m.$$

The paper determines $f(n,m)$ asymptotically for all $m$ and exactly when $m\leq t_3(n)+1$; this conjecture proposes the exact value throughout the remaining ranges of $m$.