Dicycle Cover of Hamiltonian Oriented Graphs

A dicycle cover of a digraph D is a family F of dicycles of D such that each arc of D lies in at least one dicycle in F. We investigate the problem of determining the upper bounds for the minimum number of dicycles which cover all arcs in a strong digraph. Best possible upper bounds of dicycle covers are obtained in a number of classes of digraphs including strong tournaments, Hamiltonian oriented graphs, Hamiltonian oriented complete bipartite graphs, and families of possibly non-Hamiltonian digraphs obtained from these digraphs via a sequence of 2-sum operations.

Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation

We study the qualitative behavior of the positive solutions of a second-order rational fuzzy difference equation with initial conditions being positive fuzzy numbers, and parameters are positive fuzzy numbers. More precisely, we investigate existence of positive solutions, boundedness and persistence, and stability analysis of a second-order fuzzy rational difference equation. Some numerical examples are given to verify our theoretical results.

A Note on Primitive Permutation Groups of Prime Power Degree

Primitive permutation groups of prime power degree are known to be affine type, almost simple type, and product action type. At the present stage finding an explicit classification of primitive groups of affine type seems untractable, while the product action type can usually be reduced to almost simple type. In this paper, we present a short survey of the development of primitive groups of prime power degree, together with a brief description on such groups.

On the r-Central Coefficient Matrices of the Catalan Triangles

We introduce the definition of the r-central coefficient matrices of a given Riordan array. Applying this definition and Lagrange Inversion Formula, we can calculate the r-central coefficient matrices of Catalan triangles and obtain some interesting triangles and sequences.

Product Cordial and Total Product Cordial Labelings of Pn+1m

We proved that Pn+1m is total product cordial. We also give sufficient conditions for the graph to admit (or not admit) a product cordial labeling.

On Reduced Zero-Divisor Graphs of Posets

We study some properties of a graph which is constructed from the equivalence classes of nonzero zero-divisors determined by the annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.

Radio Numbers of Certain m-Distant Trees

Radio coloring of a graph G with diameter d is an assignment f of positive integers to the vertices of G such that |f(u)-f(v)|≥1+d-d(u,v), where u and v are any two distinct vertices of G and d(u,v) is the distance between u and v. The number max {f(u):u∈V(G)} is called the span of f. The minimum of spans over all radio colorings of G is called radio number of G, denoted by rn(G). An m-distant tree T is a tree in which there is a path P of maximum length such that every vertex in V(T)∖V(P) is at the most distance m from P. This path P is called a central path. For every tree T, there is an integer m such that T is a m-distant tree. In this paper, we determine the radio number of some m-distant trees for any positive integer m≥2, and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars).

Knight’s Tours on Rectangular Chessboards Using External Squares

The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight’s tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight’s tour. In this paper we define an extended closed knight’s tour for a rectangular chessboard as a closed knight’s tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight’s tour?

Distance Degree Regular Graphs and Distance Degree Injective Graphs: An Overview

The distance d(v,u) from a vertex v of G to a vertex u is the length of shortest v to u path. The eccentricity ev of v is the distance to a farthest vertex from v. If d(v,u) = e(v), (u ≠ v), we say that u is an eccentric vertex of v. The radius rad(G) is the minimum eccentricity of the vertices, whereas the diameter diam(G) is the maximum eccentricity. A vertex v is a central vertex if e(v) = rad(G), and a vertex is a peripheral vertex if e(v) = diam(G). A graph is self-centered if every vertex has the same eccentricity; that is, rad(G) = diam(G). The distance degree sequence (dds) of a vertex v in a graph G = (V, E) is a list of the number of vertices at distance 1, 2, ... . , e(v) in that order, where e(v) denotes the eccentricity of v in G. Thus, the sequence (di0,di1,di2, …, dij,…) is the distance degree sequence of the vertex vi in G where dij denotes the number of vertices at distance j from vi. The concept of distance degree regular (DDR) graphs was introduced by Bloom et al., as the graphs for which all vertices have the same distance degree sequence. By definition, a DDR graph must be a regular graph, but a regular graph may not be DDR. A graph is distance degree injective (DDI) graph if no two vertices have the same distance degree sequence. DDI graphs are highly irregular, in comparison with the DDR graphs. In this paper we present an exhaustive review of the two concepts of DDR and DDI graphs. The paper starts with an insight into all distance related sequences and their applications. All the related open problems are listed.

Eccentric Connectivity and Zagreb Coindices of the Generalized Hierarchical Product of Graphs

Formulas for calculations of the eccentric connectivity index and Zagreb coindices of graphs under generalized hierarchical product are presented. As an application, explicit formulas for eccentric connectivity index and Zagreb coindices of some chemical graphs are obtained.