A tight lower bound on the maximum genus of a simplicial graph Original Research Article

It is proved that every connected simplicial graph with minimum valence at least three has maximum genus at least one-quarter of its cycle rank. This follows from the technical result that every 3-regular simplicial graph except K4 has a Xuong co-tree whose odd components have only one edge each. It is proved, furthermore, that this lower bound is tight. However, examples are used to illustrate that it does not apply to non-simplicial graphs. This result on maximum genus leads to several immediate consequences for average genus.