A Note on the Immersion Number of Generalized Mycielski Graphs

The immersion number of a graph G, denoted \(\textrm{im}(G)\) , is the largest t such that G has a \(K_t\) -immersion. In this note we are interested in determining the immersion number of the m-Mycielskian of G, denoted \(\mu _m(G)\) . Given the immersion number of G we provide a lower bound for \(\textrm{im}(\mu _m(G))\) . To do this we introduce the “distinct neighbor property” of immersions. We also include examples of classes of graphs where \(\textrm{im}(\mu _m(G))\) exceeds the lower bound. We conclude with a conjecture about \(\textrm{im}(\mu _m(K_t))\) .