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Edge Covers

Showing the edge cover reveals that each course will be taught by one professor and that Larry has been assigned to teach two courses.

    
Build a bipartite graph between professors and courses they are interested in teaching:
Wolfram Language code: g = [image];
Highlight the edge cover:
Wolfram Language code: colorHighlight[comp_] := HighlightGraph[g, Join@@Table[(Style[#, ColorData["Rainbow"][i / Length[comp]]])& /@ Join[comp[[i]], EdgeList[Subgraph[g, comp[[i]]]]], {i, Length[comp]}], GraphHighlightStyle -> "DehighlightHide"];
Wolfram Language code: colorHighlight[ConnectedComponents[Graph@FindEdgeCover[g]]]