public sealed class LayoutProjectItem : ArcGIS.Desktop.Internal.Core.ProjectItem, ArcGIS.Desktop.Core.IMetadata, ArcGIS.Desktop.Core.IProjectItem, System.ComponentModel.INotifyPropertyChanged  

//Reference layout project items and their associated layout.
//A layout project item is an item that appears in the Layouts folder in the Catalog pane.

//Reference all the layout project items
IEnumerable<LayoutProjectItem> layouts = Project.Current.GetItems<LayoutProjectItem>();

//Or reference a specific layout project item by name
LayoutProjectItem layoutItem = Project.Current.GetItems<LayoutProjectItem>().FirstOrDefault(item => item.Name.Equals("MyLayout"));

//Open a layout project item in a new view.
//A layout project item may exist but it may not be open in a view. 

//Reference a layout project item by name
LayoutProjectItem someLytItem = Project.Current.GetItems<LayoutProjectItem>().FirstOrDefault(item => item.Name.Equals("MyLayout"));

//Get the layout associated with the layout project item
Layout layout = await QueuedTask.Run(() => someLytItem.GetLayout());  //Worker thread

//Create the new pane
ILayoutPane iNewLayoutPane = await ProApp.Panes.CreateLayoutPaneAsync(layout); //GUI thread

//Import a pagx into a project.

//Create a layout project item from importing a pagx file
IProjectItem pagx = ItemFactory.Instance.Create(@"C:\Temp\Layout.pagx") as IProjectItem;
Project.Current.AddItem(pagx);

//This example deletes a layout in a project after finding it by name.

//Added references
using ArcGIS.Desktop.Core; 
using ArcGIS.Desktop.Layouts;

public class DeleteLayoutExample
{
  public static Task<bool> DeleteLayoutAsync(string LayoutName)
  {
    //Reference a layoutitem in a project by name
    LayoutProjectItem layoutItem = Project.Current.GetItems<LayoutProjectItem>().FirstOrDefault(item => item.Name.Equals(LayoutName));

    //Check for layoutItem
    if (layoutItem == null)
      return Task.FromResult<bool>(false);

    //Delete the layout from the project
    return Task.FromResult<bool>(Project.Current.RemoveItem(layoutItem));
  }
}