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When I right click on the grid the context menu is displayed. But the row on which I right clicked is not getting selected. I mean the CSS of the row on which I right clicked does not change. Is there any way I can highlight the right clicked row?

This is not a bug - but the correct behaviour. In many cases it is useful that selected and context menu rows are different.

 You can use setOnBeforeContextMenu event handler to assign your own function which will be called before context menu shown (you can add grid.selectCell call here for example). The version of grid which you obtain probably has a bug in this functionality. The next version will be released at the end of the week, as temporary solution just add next code to the you file after including dhtmlxgrid.js:

 dhtmlXGridObject.prototype.setOnBeforeContextMenu=function(func){

            this.dhx_attachEvent("_onBCM",func);

    };

I hope this information will be enough for you. But you also can have a look at php ui design and javascript google calendar.