Ahanjideh–Ekim–Yıldız conjecture for $Z(d)$

For odd $d$, let $Z(d)$ denote the maximum matching number of a factor-critical, almost regular, triangle-free graph of maximum degree $d$ in the class used by the paper. The paper establishes the value of $Z(d)$ for even $d$ and for $d\in\{3,5\}$, and gives bounds for the remaining odd values.

$Z(d)$ conjecture. For odd $d\geq 21$,

$$Z(d)=\left\lfloor\frac{5(d+1)}{4}\right\rfloor.$$

Determining $Z(d)$ is crucial for evaluating the proposed extremal formula and constructing the corresponding graphs. The conjecture is motivated by results suggesting that sufficiently dense triangle-free graphs are blow-ups of a $5$-cycle, but the stated range remains open in the paper.