We continue our study of online prediction of the labelling of a graph. We show a fundamental limitation of Laplacian-based algorithms: if the graph has a large diameter then the number of mistakes made by such algorithms may be proportional to the square root of the number of vertices, even when tackling simple problems. We overcome this drawback by means of an efficient algorithm which achiev...

The problem of finding a graceful labelling of a graph, or proving that the graph is not graceful, has previously been modelled as a CSP. A new and much faster CSP model of the problem has been recently proposed. This model was inspired by a mathematical proof. We present a third model that is in some sense a combination of the previous two, but is much more efficient than either. We give sever...

The bandwidth b(G) of a graph G is defined by b(G) = min max 1A(-)-X(y)1 A e={x,y} where e ranges over all edges of G and A ranges over all 1 1 functions A :V(G) -> Z , the positive integers . In this note we show for any graph G on n vertices (with G denoting its complement), b(G) + b(G) > n 2 Furthermore, for all n > 3 there exist graphs which achieve this bound . We also prove : (i) b(G) + b...

An orthogonal double cover (ODC) of a graph H is a collection ( ) { } v G v V H : = ∈  of ( ) V H subgraphs (pages) of H, so that they cover every edge of H twice and the intersection of any two of them contains exactly one edge. An ODC  of H is cyclic (CODC) if the cyclic group of order ( ) V H is a subgroup of the automorphism group of  . In this paper, we introduce a general orthogonal la...

Graphs and graph transformations are natural means to describe systems of processes. Graphs represent structure of the system, and graph rewriting rules model dynamic behaviour. In this chapter, we illustrate the technique by describing Petri nets, statecharts, parallel logic programming, and systems of processes. Whereas description of Petri nets is based on usual graphs, statecharts lead us t...

Let j ≥ k ≥ 0 be integers. An `-L(j, k)-labelling of a graph G = (V , E) is a mapping φ : V → {0, 1, 2, . . . , `} such that |φ(u)−φ(v)| ≥ j if u, v are adjacent and |φ(u)−φ(v)| ≥ k if they are distance two apart. Let λj,k(G) be the smallest integer ` such that G admits an `-L(j, k)-labelling. Define λj,k(G) to be the smallest ` if G admits an `-L(j, k)-labelling with φ(V ) = {0, 1, 2, . . . , ...

A graph G is distinguished if its vertices are labelled by a map φ : V (G) −→ {1, 2, . . . , k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ : X −→ {1, 2, . . . , ...

Sense of direction is a property of the labelling of (possibly anonymous) networks which allows to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. A graph has minimal sense of direction whenever it has sense of direction and the number of colours equals its maximum outdegree. We prove that an outregular digraph with minimal weak s...

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Motivated by problems in radio channel assignment, we consider the vertexlabelling of graphs with nonnegative integers. The objective is to minimize the span of the labelling, subject to constraints imposed at graph distances one and two. We show that the minimum span is (up to rounding) a piecewise linear function of the constraints, and give a complete specification, together with the associa...