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We propose to use HII galaxies (HIIG) to trace the redshift-distancerelation, by means of their $L(\mathrm{H}\beta) - \sigma$ correlation, in anattempt to constrain the dark energy equation of state parameter solutionspace, as an alternative to the cosmological use of type Ia supernovae.  For a sample of 128 local compact HIIG with high equivalent widths of theirBalmer emission lines we obtained ionised gas velocity dispersion from highS/N, high-dispersion spectroscopy (Subaru-HDS and ESO VLT-UVES) and integratedH$\beta$ fluxes from low dispersion wide aperture spectrophotometry. We findthat the $L(\mathrm{H}\beta) - \sigma$ relation is strong and stable againstrestrictions in the sample. The size of the starforming region is an importantsecond parameter, while adding the emission line equivalent width or thecontinuum colour and metallicity, produces the solution with the smallest rmsscatter.  We have used the $L(\mathrm{H}\beta) - \sigma$ relation from a local sampleof HIIG and a local calibration or `anchor', given by giant HII regions innearby galaxies which have accurate distance measurements determined viaprimary indicators, to obtain a value of $H_0$. Using our best sample of 69HIIG and 23 Giant HII regions in 9 galaxies we obtain $H_{0}=74.3 \pm 3.1$(statistical)$\pm$ 2.9 (systematic) km s$^{-1}$ Mpc$^{-1}$, in excellentagreement with, and independently confirming, the most recent SNa Ia basedresults.  Using a local sample (107 sources) and a sample of 21 high redshift HIIG, 6of them with medium-dispersion spectroscopy (ESO VLT-XShooter) and 17 takenfrom the literature, we have obtained constraints on the planes $H_0 -\Omega_m$, $\Omega_m - w_0$ and $w_0 - w_1$ (CPL model).

Constraining the Parameter Space of the Dark Energy Equation of State Using Alternative Cosmic Tracers