A Family of Graceful Diameter Six Trees Generated by Component Moving Techniques

Aims/ Objectives: To identify some new classes of graceful diameter six trees using component moving transformation techniques.
Study Design: Literature Survey to our findings.
Place and Duration of Study: Department of Mathematics, C.V. Raman College of Engineering, Bhubaneswar, India, between June 2014 and September 2016.
Methodology: Component Moving Transformation.
Results: Here a diameter six tree is denoted by (a0; a1; a2; ....; am; b1; b2; ....; bn; c1; c2; ....; cr) with a0 as the center of the tree, ai; i = 1; 2; ....;m, bj ; j = 1; 2; ....; n, and ck; k = 1; 2; ....; r are the vertices of the tree adjacent to a0; each ai is the center of some diameter four tree, each bj is the center of some star, and each ck is some pendant vertex. This article gives graceful labelings to a family of diameter six trees (a0; a1; a2; ....; am; b1; b2; ....; bn; c1; c2; ....; cr) with diameter four trees incident on ais possess an odd number of branches comprising of six different combinations of odd, even, and pendant branches. Here a star is called an odd branch if its center has an even degree, an even branch if its center has an odd degree, and a pendant branch if its center has degree one.

Conclusions: Our article finds many new graceful diameter six trees by component moving techniques. However, the problem that all diameter six trees are graceful is still open and we conclude that one can not give graceful labelings to all diameter six trees by component moving techniques.