Janusz Dybizbański

Hamiltonian paths in hypercubes with local traps

We consider hypercube Qn with one or two vertices of degree one.In such cubes some pairs of vertices can be connected by Hamiltonian paths.We show that they can be connected even if n - 3 additional edges are faulty. The n-dimensional hypercube Qn is a graph with 2n vertices, each labeled with a distinct binary string of length n. The vertices are connected by an edge if and only if their labels differ in one bit. The hypercube is bipartite, the set of nodes is the union of two sets: nodes of parity 0 (the number of ones in their labels is even) and nodes of parity 1 (the number of ones is odd). We consider Hamiltonian paths in hypercubes with faulty edges and prove the following: (1) If Qn has one vertex u of degree 1, then u can be connected by a Hamiltonian path with every vertex v that is of a parity different than u and that is not connected with u by a healthy edge. (2) If Qn with n ? 4 has two vertices u and v of degree 1, then they can be connected by a Hamiltonian path if the distance between u and v is odd and greater than 1 or if u and v are connected by the faulty edge. (3) If Qn contains a cycle (u, v, w, x) in which all edges going away from the cycle from u and w are faulty, then u or w can be connected by a Hamiltonian path with any vertex outside the cycle that is of different parity than u and w.Moreover, in all three cases, the thesis remains true even if Qn has n - 3 additional faulty edges. Furthermore, in all three cases, no other Hamiltonian paths are possible.

The oriented chromatic number of Halin graphs

The oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number of an unoriented graph G is the maximal chromatic number over all possible orientations of G. In this paper, we prove that every Halin graph has oriented chromatic number at most 8, improving a previous bound by Hosseini Dolama and Sopena, and confirming the conjecture given by Vignal. We prove that every oriented Halin graph can be colored with at most 8 colors.Hence, the oriented chromatic number of the family of Halin graphs is equal to 8.This confirms the conjecture given by Vignal and solves an open problem presented by Sopena.

Hamiltonian cycles in hypercubes with faulty edges

ABSTRACT Szepietowski [12] observed that the hypercube is not Hamiltonian if it contains a trap disconnected halfway. A proper subgraph T is disconnected halfway if at least half of its nodes have parity 0 (or 1, resp.) and all edges joining the nodes of parity 0 (or 1, resp.) in T with nodes outside T, are faulty. In this paper, we describe all traps disconnected halfway T with the size and we consider the problem whether there exist small sets of faulty edges that are not based on sets disconnected halfway and still preclude Hamiltonian cycles. We show that if with the set of faulty edges F contains a trap disconnected halfway T that is of size and is minimal, then T is a path or is Hamiltonian. We also describe heuristic that recognizes nonhamiltonian cubes, also these ones that do not contain traps disconnected halfway.

Oriented chromatic number of Cartesian products and strong products of paths

Abstract An oriented coloring of an oriented graph G is a homomorphism from G to H such that H is without selfloops and arcs in opposite directions. We shall say that H is a coloring graph. In this paper, we focus on oriented col- orings of Cartesian products of two paths, called grids, and strong products of two paths, called strong-grids. We show that there exists a coloring graph with nine vertices that can be used to color every orientation of grids with five columns. We also show that there exists a strong-grid with two columns and its orientation which requires 11 colors for oriented coloring. Moreover, we show that every orientation of every strong-grid with three columns can be colored by 19 colors and that every orientation of every strong-grid with four columns can be colored by 43 colors. The above statements were proved with the help of computer programs.

On some three-color Ramsey numbers for paths

For graphs G 1 , G 2 , G 3 , the three-color Ramsey number R ( G 1 , G 2 , G 3 ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it contains a monochromatic copy of G i in color i , for some 1 ? i ? 3 .First, we prove that the conjectured equality R ( C 2 n , C 2 n , C 2 n ) = 4 n , if true, implies that R ( P 2 n + 1 , P 2 n + 1 , P 2 n + 1 ) = 4 n + 1 for all n ? 3 . We also obtain two new exact values R ( P 8 , P 8 , P 8 ) = 14 and R ( P 9 , P 9 , P 9 ) = 17 , furthermore we do so without help of computer algorithms. Our results agree with a formula R ( P n , P n , P n ) = 2 n - 2 + ( n mod 2 ) which was proved for sufficiently large n by Gyarfas, Ruszinko, Sarkozy, and Szemeredi (2007). This provides more evidence for the conjecture that the latter holds for all n ? 1 .