On the Average Crosscap Number of a Graph

Abstract

The average crosscap number of a graph \(G\) is the expected value of the crosscap number random variable, over all labeled \(2\)-cell non-orientable embeddings of \(G\). In this study, some experimental results for average crosscap number are obtained. We calculate all average crosscap numbers of graphs with Betti number less than \(5\). As a special case, the smallest ten values of average crosscap number are determined. The distribution of average crosscap numbers of all graphs in \({R}\) is sparse. Some structure theorems for average crosscap number with a given or bounded value are provided. The exact values of average crosscap numbers of cacti and necklaces are determined. The crosscap number distributions of cacti and necklaces of type \((r,0)\) are proved to be strongly unimodal, and the mode of the embedding distribution sequence is upper-rounding or lower-rounding of its average crosscap number. Some open problems are also proposed.