The Gray-code Graph of t-ary Trees Using RD-sequences Is Hamiltonian∗

Usually, t-ary trees are encoded to integer sequences as their representation. It has been shown that the right-distance sequences (or RD-sequence for short) are concise representations of t-ary trees and are suitable for the usage of loopless Gray-code enumeration. For n, t 2, the RD-representation graph, denoted as RDt n, is a graph whose vertices are the RD-sequences of t-ary trees with n internal nodes, and two vertices are joined by an edge if and only if their RD-sequences differ in exactly one position. From the definition, a Gray-code enumeration for t-ary trees can be implemented by finding a Hamiltonian path in RDt n. In [24], the authors have proposed an algorithm to find a Hamiltonian path in RDt n. In this paper, we further determine some basic graph parameters of RDt n, including order, size, degrees of vertices, eccentricity, diameter, radius, center, and median. Moreover, we show that every RD-representation graph is Hamiltonian, i.e., it contains a simple cycle passing through all vertices of RDt n. ∗This research was partially supported by National Science Council under the Grants NSC95-2221-E-260024. 1 . Introduction An easy way to represent combinatorial objects is the use of integer sequences encoding. To generate objects efficiently, enumerating algorithms are postulated to run in a constant time for each generation under the worst case, and so that the number of changes between two consecutive sequences can be bounded by a constant. A recommendable choice for designing an enumerating algorithm is the use of a technique called loopless (i.e., algorithm involves no recursion and, after initialization, uses only “if-then-else” statements and assignments) [2, 23] and such that the output sequences result in a Gray-code order [18]. A Gray-code graph of objects is a graph containing all objects of the same size as its vertices and such that two vertices are connected by an edge if their corresponding sequences differ in exactly one position. Consequently, an implementation of Gray-code enumeration for objects can be done by finding a Hamiltonian path in the Gray-code graph. Till now, many researches have been devoted to the loopless Gray-code enumeration of biThe 24th Workshop on Combinatorial Mathematics and Computation Theory