We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schel...

The traveling salesman problem (TSP) is one of the most fundamental problems in combinatorial optimization. Given a graph, the goal is to find a Hamiltonian cycle of minimum or maximum weight. We consider finding Hamiltonian cycles of maximum weight (Max-TSP). An instance of Max-TSP is a complete graph G = (V,E) with edge weights w : E → N. The goal is to find a Hamiltonian cycle of maximum wei...

A k−uniform hypergraphH is a pair (V, ε), where V = {v1, v2, . . . , vn} is a set of n vertices and ε is a family of k-subset of V called hyperedges. A cycle of length l of H is a sequence of the form (v1, e1, v2, e2, . . . , vl, el, v1), where v1, v2, . . . , vl are distinct vertices, and e1, e2, . . . , el are k-edges of H and vi, vi+1 ∈ ei, 1 ≤ i ≤ l, where addition on the subscripts is modu...

A. S. Asratian DEPARTMENT OF MATHEMATICS UNIVERSITY OF UMEA S-907 87, UMEA, SWEDEN DEPARTMENT OF MATHEMATICAL CYBERNETICS YEREVAN STATE UNIVERSITY YEREVAN, 375049, REPUBLIC OF ARMENIA R. Haggkvist DEPARTMENT OF MATHEMATICS UNIVERSITY OF UMEA S-907 87, UMEA, SWEDEN G. V. Sarkisian DEPARTMENT OF MATHEMATICAL CYBERNETICS YEREVAN STATE UNIVERSITY YEREVAN, 375049, REPUBLIC OF ARMENIA It is well know...

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schel...

1 I n t r o d u c t i o n All graphs in this paper are connected and simple, i.e. finite, undirected, loopless and without multiple edges. In a graph G = (V, E) the length of a path from a vertex v to a vertex u is the number of edges in the path. The distance d(u, v) from vertex u to vertex v is the length of a minimum length path from u to v and the interval I(u, v) between these vertices is ...

PURPOSE The relationship between unhealthy body mass index (BMI) and adherence to orthodontic treatment with removable appliances has not previously been evaluated. OBJECTIVE The aim of this study was to quantify the association between BMI and wear time of removable orthodontic appliances and to evaluate BMI changes during orthodontic treatment. PATIENTS AND METHODS Fifty-three normal-weig...

We show that the 4-coloring problem can be solved in polynomial time for graphs with no induced 5-cycle C5 and no induced 6-vertex path P6.

In some recent papers the method of partial differences introduced by the author in [4] was very helpful in the construction of cyclic cycle systems. Here we use and describe in all details this method for the more general purpose of constructing cycle decompositions with a sharply vertex transitive automorphism group not necessarily cyclic.

The vertex arboricity a(G) of a graph G is the minimum k such that V (G) can be partitioned into k sets where each set induces a forest. For a planar graph G, it is known that a(G) ≤ 3. In two recent papers, it was proved that planar graphs without k-cycles for some k ∈ {3, 4, 5, 6, 7} have vertex arboricity at most 2. For a toroidal graph G, it is known that a(G) ≤ 4. Let us consider the follo...