Saad El-Zanati

On the Hamilton-Waterloo problem

Abstract. The Hamilton-Waterloo problem asks for a 2-factorisation of Kv in which r of the 2-factors consist of cycles of lengths a1,a2,…,at and the remaining s 2-factors consist of cycles of lengths b1,b2,…,bu (where necessarily ∑i=1tai=∑j=1ubj=v). In this paper we consider the Hamilton-Waterloo problem in the case ai=m, 1≤i≤t and bj=n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}.

On Rosa-type labelings and cyclic graph decompositions

A labeling (or valuation) of a graph G is an assignment of integers to the vertices of G subject to certain conditions. A hierarchy of graph labelings was introduced by Rosa in the late 1960s. Rosa showed that certain basic labelings of a graph G with n edges yielded cyclic G-decompositions of K2n+1 while other stricter labelings yielded cyclic G-decompositions of K2nx+1 for all natural numbers x. Rosa-type labelings are labelings with applications to cyclic graph decompositions. We survey various Rosa-type labelings and summarize some of the related results.

Star Decompositions of Cubes

Abstract. Let Sk denote the complete bipartite graph K1,k and let Qn denote the n-cube. We prove that the obvious necessary conditions for the existence of an Sk-decomposition of Qn are sufficient.

Factorizations of complete multipartite graphs into generalized cubes

A graph is t-tough if the number of components of G\S is at most |S|-t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a Hamilton cycle, and a longstanding conjecture by Chvatal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k-walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4-tough) graph has a 2-walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k-tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k-walk for k ≥ 2.

Maximum packings with odd cycles

Abstract We show how to obtain maximum packings of K 2 kg + v with k -cycles when k ⩾3 is odd, g a positive integer, and v even with 0⩽ v k . Moreover, under certain conditions on v , we obtain maximum packings of K 2 kg + v .

On Cyclic Decompositions of Complete Graphs into Tripartite Graphs

We introduce two new labelings for tripartite graphs and show that if a graph G with n edges admits either of these labelings, then there exists a cyclic G-decomposition of for every positive integer x. We also show that if G is the union of two vertext-disjoint cycles of odd length, other than , then G admits one of these labelings.

Maximal sets of Hamilton cycles in K n,n

A graph G is perfectly orderable, if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph, if it contains no P4, 2K2, and C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev, and de Werra states that a graph is perfectly orderable, if it is the union of two threshold graphs. In this article, we investigate possible generalizations of the above theorem. Hoang has conjectured that, if G is the union of two graphs G1 and G2, then G is perfectly orderable whenever G1 and G2 are both P4-free and 2K2-free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to this conjecture, and we prove a special case of it: if G1 and G2 are two edge-disjoint graphs that are P4-free and 2K2-free, then the union of G1 and G2 is perfectly orderable.

Factorizations of Km,n into Spanning Trees

Abstract. We establish necessary and sufficient conditions for the existence of a spanning tree factorization of the complete bipartite graph Km,n.

On dynamics of certain Cantor sets

Abstract We answer the following question: If p and q are positive integers greater than 1 and C p is the set of all numbers in [0, 1] which can be expressed in base p without using a nonempty finite collection of finite length patterns in Z p , under what conditions does C p contain a number whose base q expansion contains all patterns of finite length?

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.