Alternating Graphs

In this paper we generalize the concept of alternating knots to alternating graphs and show that every abstract graph has a spatial embedding that is alternating. We also prove that every spatial graph is a subgraph of an alternating graph. We define n-composition for spatial graphs and generalize the results of Menasco on alternating knots to show that an alternating graph is n-composite for n=0, 1, 2, 3 if and only if it is “obviously n-composite” in any alternating projection. Moreover, no closed incompressible pairwise incompressible surface exists in the complement of an alternating graph. We then generalize results of Kauffman, Murasugi, and Thistlethwaite to prove that the crossing number of an even-valent rigid-vertex alternating spatial graph is realized in every reduced alternating projection with no uncrossed cycles and, if the graph is not 2-composite, the crossing number is not realized in any non-alternating projection. We give examples showing that this result does not hold for graphs with vertices of odd valence or graphs with uncrossed cycles.