We investigate the following very simple load-balancing algorithm on the N-cycle (N even) which we call Odd-Even Transposition Balancing (OETB). The edges of the cycle are partitioned into two matchings canonically. A matching defines a single step, the two matchings form a single round. Processors connected by an edge of the matching perfectly balance their loads, and, if there is an excess to...

In this paper we present an O( I V I + I E I) time algorithm which partitions the set of vertices of an undirected graph G = (I’, E) with maximum degree A and minimum degree 6 to {A, x} such that the following hold. (8 The number of edges with one end point in A and the other end point in x is at least IEl/2. (ii> The difference between the number of vertices in A and the number of vertices in ...

Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

We will characterize all graphs that have the property that the graph and its complement are minimal even pair free. This characterization allows a new formulation of the Strong Perfect Graph Conjecture. The reader is assumed to be familiar with perfect graphs (see for example [2]). A hole is a cycle of length at least five. An odd hole is a hole that has an odd number of vertices. An (odd) ant...

Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

Girth pairs were introduced by Harary and Kovács [Regular graphs with given girth pair, J. Graph Theory 7 (1983) 209–218]. The odd girth (even girth) of a graph is the length of a shortest odd (even) cycle. Let g denote the smaller of the odd and even girths, and let h denote the larger. Then (g, h) is called the girth pair of the graph. In this paper we prove that a graph with girth pair (g, h...