On Normal q-Ary Codes in Rosenbloom-Tsfasman Metric

The notion of normality of codes in Hamming metric is extended to the codes in Rosenbloom-Tsfasman metric (RT-metric, in short). Using concepts of partition number and l-cell of codes in RT-metric, we establish results on covering radius and normality of q-ary codes in this metric. We also examine the acceptability of various coordinate positions of q-ary codes in this metric. And thus, by exploring the feasibility of applying amalgamated direct sum method for construction of codes, we analyze the significance of normality in RT-metric.

Noncontiguous Pattern Containment in Binary Trees

We consider the enumeration of binary trees containing noncontiguous binary tree patterns. First, we show that any two ℓ-leaf binary trees are contained in the set of all n-leaf trees the same number of times. We give a functional equation for the multivariate generating function for number of n-leaf trees containing a specified number of copies of any path tree, and we analyze tree patterns with at most 4 leaves. The paper concludes with implications for pattern containment in permutations.

Some New Results on Prime Cordial Labeling

A prime cordial labeling of a graph G with the vertex set V(G) is a bijection f:V(G)→{1,2,3,…,|V(G)|} such that each edge uv is assigned the label 1 if gcd(f(u),f(v))=1 and 0 if gcd(f(u),f(v))>1; then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits a prime cordial labeling is called a prime cordial graph. In this work we give a method to construct larger prime cordial graph using a given prime cordial graph G. In addition to this we have investigated the prime cordial labeling for double fan and degree splitting graphs of path as well as bistar. Moreover we prove that the graph obtained by duplication of an edge (spoke as well as rim) in wheel Wn admits prime cordial labeling.

The Diophantine Equation 8x2-y2+8x(1+t)+(2t+1)2=0 and t-Balancing Numbers

Let t≥2 be an odd integer such that 2t2-1 is a prime. In this work, we determine all integer solutions of the Diophantine equation Dt:8x2-y2+8x(1+t)+(2t+1)2=0 and then we deduce the general terms of all t-balancing numbers.

Integer Semigroups Associated with Dumont-Thomas Numeration Systems

Given a primitive substitution, we define different binary operations on infinite subsets of the nonnegative integers. These binary operations are defined with the help of the Dumont-Thomas numeration system; that is, a numeration system associated with the substitution. We give conditions for these semigroups to have an identity element. We show that they are not finitely generated. These semigroups define actions on the set of positive integers. We describe the orbits of these actions. We also estimate the density of these sets as subsets of the positive integers.

An Extension of a Congruence by Tauraso

For a positive integer let be the th harmonic number. In this paper we prove that, for any prime , . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.

The -Version of Binary Search Trees: An Average Case Analysis

Following a suggestion of Cichoń and Macyna, binary search trees are generalized by keeping (classical) binary search trees and distributing incoming data at random to the individual trees. Costs for unsuccessful and successful search are analyzed, as well as the internal path length.

Partitions of Natural Numbers with the Intersection Not Empty

Let ℕ be the set of nonnegative integers. For a given set A⊂ℕ the representation functions R2(A,n), R3(A,n) are defined as the number of solutions of the equation n=a+a′, a,a′∈A with condition a<a′, a≤a′, respectively. In this paper, we prove that if ℕ=A∪B and A∩B={8k:k∈ℕ}, then Ri(A,n)=Ri(B,n) cannot hold for all sufficiently large integers n where i=2,3.

-Tuple Total Restrained Domination in Complementary Prisms

In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.

Packings and Coverings of Various Complete Graphs with the 4-Cycle with a Pendant Edge

We consider the packings and coverings of complete graphs with isomorphic copies of the 4-cycle with a pendant edge. Necessary and sufficient conditions are given for such structures for (1) complete graphs , (2) complete bipartite graphs , and (3) complete graphs with a hole . In the last two cases, we address both restricted and unrestricted coverings.