Noncrossing Trees and Noncrossing Graphs We give a parity reversing involution on noncrossing trees that leads to a combinatorial interpretation of a formula on noncrossing trees and symmetric ternary trees in answer to a problem proposed by Hough.  We use the representation of Panholzer and Prodinger for noncrossing trees and find a correspondence between a class of noncrossing trees, called proper noncrossing trees, and the set of symmetric ternary trees. The second result of this paper is a parity reversing involution on connected noncrossing graphs which leads to a relation between the number of noncrossing trees with n edges and k descents and the number of connected noncrossing graphs with n1 vertices and m edges.

On the Crossing Numbers of the Joining of a Specific Graph on Six Vertices with the Discrete Graph In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product      G *    D n      for the graph     G *     on six vertices consists of one vertex which is adjacent with three nonconsecutive vertices of the 5cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph     G *     exactly once. These minimum symmetrical values are described in the individual symmetric tables.