On the Crossing Numbers of Cartesian Products of Stars and Graphs of Order Six

The crossing number $ \text{cr}(G) $ of a graph $ G $ is the minimal number of crossings over all drawings of $ G $ in the plane. According to their special structure, the class of Cartesian products of two graphs is one of few graph classes for which some exact values of crossing numbers were obtained. The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. Moreover, except of six graphs, the crossing numbers of Cartesian products $ G \square K_{1,n} $ for all other connected graphs $ G $ on five vertices are known. In this paper we are dealing with the Cartesian products of stars with graphs on six vertices. We give the exact values of crossing numbers for some of these graphs and we summarise all known results concerning crossing numbers of these graphs. Moreover, we give the crossing number of $ G_1 \square T $ for the special graph $ G_1 $ on six vertices and for any tree $ T $ with no vertex of degree two as well as the crossing number of $ K_{1,n} \square T $ for any tree $ T $ with maximum degree five.