Abdollah Khodkar

3,5-Cycle decompositions

For all odd integers n and all non-negative integers r and s satisfying 3r + 5s = n(n -1)/2 it is shown that the edge set of the complete graph on n vertices can be partitioned into r 3-cycles and s 5-cycles. For all even integers n and all non-negative integers r and s satisfying 3r + 5s = n(n-2)/2 it is shown that the edge set of the complete graph on n vertices with a 1-factor removed can be partitioned into r 3-cycles and s 5-cycles

On Alspach's conjecture with two even cycle lengths

Abstract For m,n even and n>m , the obvious necessary conditions for the existence of a decomposition of the complete graph K v when v is odd (or the complete graph with a 1-factor removed K v ⧹F when v is even) into r m -cycles and s n -cycles are shown to be sufficient if and only if they are sufficient for v . This result is used to settle all remaining cases with m,n⩽10 .

Steiner Trades That Give Rise To Completely Decomposable Latin Interchanges

In this paper we focus on the representation of Steiner trades of volume less than or equal to nine and identify those for which the associated partial latin square can be decomposed into six disjoint latin interchanges.

(m,n)-cycle systems

We describe a method which, in certain circumstances, may be used to prove that the well-known necessary conditions for partitioning the edge set of the complete graph on an odd number of vertices (or the complete graph on an even number of vertices with a 1-factor removed) into cycles of lengths m(1),m(2),...,m(t) are sufficient in the case \{m(1), m(2), ..., m(t)}\=2. The method is used to settle the case where the cycle lengths are 4 and 5

Global signed total domination in graphs

A function f : V (G) --> {-1, 1} defined on the vertices of a graph G is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. A STDF f of G is called a global signed total dominating function (GSTDF) if f is also a STDF of the complement G of G. The global signed total domination number ygst(G) of G is defined as ygst(G) = minf {sumatorio vEV (G) f(v) / f is a GSTDF of G}. In this paper first we find lower and upper bounds for the global signed total domination number of a graph. Then we prove that if T is a tree of order n > 4 with A(T) < n - 2, then Ygst(T) < Yst(T) + 4. We characterize all the trees which satisfy the equality. We also characterize all trees T of order n >= 4, A(T) <= n - 2 and Ygst(T) = Yst(T) + 2.

The m -way intersection problem for m -cycle systems

An m-cycle system of order upsilon is a partition of the edge-set of a complete graph of order upsilon into m-cycles. The mu -way intersection problem for m-cycle systems involves taking mu systems, based on the same vertex set, and determining the possible number of cycles which can be common to all mu systems. General results for arbitrary m are obtained, and detailed intersection values for (mu, m) = (3, 4), (4, 5),(4, 6), (4, 7), (8, 8), (8, 9). (For the case (mu, m)= (2, m), see Billington (J. Combin. Des. 1 (1993) 435); for the case (Cc,m)=(3,3), see Milici and Quattrochi (Ars Combin. A 24 (1987) 175

On a generalization of the Oberwolfach problem

Let e1, e2, ..., en be a sequence of nonnegative integers such that the first non-zero term is not one. Let Σi - 1n ei = (q - 1)/2, where q = pn and p is an odd prime. We prove that the complete graph on q vertices can be decomposed into e1 Cpn-factors, e2 Cpn - 1-factors, ..., and en Cp-factors.

On Orthogonal Double Covers of Graphs

A transformation which allows us to obtain an orthogonal double cover of a graph G from any permutation of the edge set of G is described. This transformation is used together with existence results for self-orthogonal latin squares, to give a simple proof of a conjecture of Chung and West.

Uniquely Bipancyclic Graphs

In this chapter we look at the problem of uniquely bipancyclic graphs, that is bipartite graphs that contain exactly one cycle of each length from 4 up to the number of vertices. If such a graph contains c cycles, their lengths are 4, 6, …, 2c + 2, so the graph has 2c + 2 vertices; moreover the sum of the lengths of the cycles (total number of edges in the cycles, with multiple appearances in different cycles counted multiply) is \(4 + 6 + \cdots + (2c + 2) = c^{2} + 3c\). We shall denote this by s(c).

Difference Covering Arrays and Pseudo-Orthogonal Latin Squares

A pair of Latin squares, A and B, of order n, is said to be pseudo-orthogonal if each symbol in A is paired with every symbol in B precisely once, except for one symbol with which it is paired twice and one symbol with which it is not paired at all. A set of t Latin squares, of order n, are said to be mutually pseudo-orthogonal if they are pairwise pseudo-orthogonal. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.