I'm not familiar with a way to do that inside Wolfram Alpha itself, which I think is what you're after. But since you found the forum for the Wolfram language... just in case you're curious, an example of how you could do it via Wolfram programming (if you haven't already learned Wolfram programming). What's described below would be done in a Wolfram programming environment (e.g. Wolfram Development Platform), not Wolfram Alpha itself.

To get a list of all the graph classes available (including Eulerian, Planar, etc):

availableGraphClasses = GraphData["Classes"]

To list all the properties you can display for a graph:

availableProperties = Complement @@ GraphData /@ {"Properties", "Classes"}

The graph classes you want all the search results to belong to:

graphClasses = {"Nonhamiltonian", "Eulerian", "PerfectMatching", 
"Planar"};

Retrieve the graphs that belong to all the classes you've specified:

searchResults = Intersection @@ Map[GraphData[#] &, graphClasses]

Put it all together, and generate a viewer to display whatever properties you want for the graphs retrieved:

propertiesToDisplay = {"AdjacencyMatrix", "AllImages", 
"CanonicalForm", "Center", 
"EdgeCount", "EdgeRules", "Information", "StandardNames", 
"VertexCount"};
graphClasses = {"Nonhamiltonian", "Eulerian", "PerfectMatching", 
"Planar"};
searchResults = Intersection @@ Map[GraphData[#] &, graphClasses]
Manipulate[Column[{GraphData[g], GraphData[g, p]}], {g, 
graphSearchResults}, {p, propertiesToDisplay}]

Example viewer, with just a few properties displayed, following examples in documentation

Anyways, I know you're probably looking for a purely Wolfram Alpha solution, but I just thought I'd toss that out there in case you were curious. Best wishes...

https://www.wolframcloud.com/objects/1833d555-f964-4a02-96d2-bf6e19a50ea7