On hamiltonian decomposition of direct graph bundle Irena Hrastnik Ladinek and Janez Žerovnik Abstract—Hamiltonian decomposition of direct graph bundles is studied. Based on the recent proof of hamiltonicity of all con- nected direct graph bundles over hamiltonian base and hamiltonian fibres, we conjecture that all direct graph bundles with fibres and base graphs being hamiltonian decomposable also admit a hamil- tonian decomposition. The conjecture is proved for direct bundles over cycles when the nontrivial automorphism is any reflection. We also prove that direct graph bundles with α a cyclic shift when the base cycle is even admit a hamiltonian decomposition. In the case of odd base cycle we look at one partical situation where we can construct a hamiltonian decomposition. Keywords—circulant 2-digraph, cyclic ‘-shift, direct graph product, direct graph bundle, hamiltonian graph, hamiltonian de- composition, reflection. I. INTRODUCTION S TUDIES of hamiltonian properties of graphs are among the fundamental topics in graph theory [10],[24]. Besides being related to some famous historical problems (Icosian game, chessboard puzzles, etc.) it has important practi- cal applications. For example, in computer science, hamil- tonicity and existence of hamiltonian decomposition are im- portant properties of computer and communication network topologies. Furthermore, the traveling salesman problem [32] which is the most studied problem in combinatorial optimization asks for a minimal hamiltonian cycle in edge weighted graph. There is no efficient algorithm for deciding whether a graph is hamiltonian or not. (More precisely, as the problem is NP-complete, it is believed that there is no poly- nomial algorithm.) Therefore it is interesting to ask, given a subclass of graphs, whether the problem may be solved efficiently by designing a polynomial algorithm or by pro- viding a characterization of hamiltonian graphs within the subclass. Graph products are one of the natural construc- tions giving more complex graphs from simple ones. Graph Irena Hrastnik Ladinek is with the Faculty of Mechanical Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia, (correspond- ing author, e-mail: irena.hrastnik@um.si). Janez Žerovnik is with the Faculty of Mechanical Engineering, Univer- sity of Ljubljana, Aškerˇ ceva 6, 1000 Ljubljana and with the Istitute of Math- ematics, Physics, and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia, (e-mail:janez.zerovnik@imfm.si). bundles, sometimes also called twisted products, are a gene- ralization of product graphs, which have been (under various names) frequently used as computer topologies or communi- cation networks, see for example [6]. A famous example is the ILIAC IV supercomputer [8]. While hamiltonian proper- ties of the cartesian products are well studied, there is much less known on hamiltonian properties of direct products and bundles. The reason may be that the direct product has some, on the first sight not convenient properties. For example, the direct product of connected graphs is not necessarily con- nected. The direct product is one of the (four) most important graph products. It is in some sense the most natural graph product as it can be viewed as the product in the category of graphs. The product was used by Greenwell and Lovasz [21] to demonstrate that for all n ≤ 3, there is a uniquely n-colorable graph without odd cycles shorter than a given number s. Whether a product of hamiltonian decomposable graphs is itself hamiltonian decomposable has been an object of study for a long time. For example, Barayani and Szasz [7] showed that this problem admits of an affirmative answer with respect to the lexicographic product. Jha [22] proved that if the number of factor graphs of even order is at most one, then the direct product admits a hamiltonian decompo- sition and, if the number of factor graphs which are bipartite is at least two and the remaining factor graphs are all of odd order, then the direct product consists of isomorphic compo- nents each of which admits a hamiltonian decomposition. Our less general motivation for this research is the fol- lowing. It is well-known that the Cartesian product of two hamiltonian graphs is hamiltonian, and therefore it is inter- esting to investigate conditions under which the product is hamiltonian if at least one of the factors is not hamiltonian. In 1982, Batagelj and Pisanski [9] proved that the Cartesian product of a tree T and a cycle C n has a hamiltonian cycle if and only if n ≥ Δ(T ), where Δ(T ) denotes the maximum vertex degree of T . They introduced the cyclic hamiltonic- ity cH(G) of graph G as the smallest integer n for which the Cartesian product of cycle C n and G is hamiltonian. More than twenty years later, Dimakopoulos, Palios and Paulaki- das [14] proved that cH(G) ≤ D (G) ≤ cH(G)+ 1, as con- jectured already in [9]. (Here D (G) denotes the minimum 1 INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 10, 2016 ISSN: 1998-0140 120