In this paper we introduce the “do not touch” condition for squares in the discrete plane. We say that two squares “do not touch” if they do not share any vertex or any segment of an edge. Using this condition we define a covering of the discrete plane, the covering can be strong or weak, regular or non-regular. For simplicity, in this article, we will restrict our attention to regular covering...

The problem of nding the minimum topology of multipro-cessing substrates supporting parallel execution of any given neighbourhood constrained system is proposed and possible optimal strategies are investigated based on the relationship between Barbosa's scheduling by edge reversal-SER-distributed algorithm and the minimum clique covering problem. It is shown that from any given clique covering ...

Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization matchings branchings. In this paper, we propose edge cover covering counterpart forest, extend matching–covering framework graphs. While result...

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degre...

For a finite connected graph let be the spectral radius of its universal cover. We prove that for any graph of average degree and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph after deleting any radius ball is at least , then its second largest eigenvalue in absolute value is at least for some absolute constant . This result is tight i...

The main theorem states that if K is a finite CW-complex with finite fundamental group G and universal cover homotopy equivalent to a product of spheres X, then G acts smoothly and freely on X×Sn for any n greater than or equal to the dimension of X. If the G-action on the universal cover of K is homologically trivial, then so is the action on X × Sn. Ünlü and Yalçın recently showed that any fi...

A 2-graphoidal cover of a graph G is a collection ψ of paths (not necessarily open) in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most two paths in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a 2-graphoidal cover of G is called the 2-graphoidal covering number of G and is denoted by η2(G) or η2. Here, we...

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip3 or the complete graph K4, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of...

If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a ...