In this article we propose a new model for scheduling periodic tasks. The model is based on a variation of the circular chromatic number, called the multiple circular colouring of the conflict graph. We show that for a large class of graphs, this new model will provide better solutions than the original circular chromatic number. At the same time, it allows us to avoid the difficulty of impleme...

McDiarmid C.J.H. and A. Sanchez-Arroyo, An upper bound for total colouring of graphs, Discrete Mathematics 111 (1993) 3899392. We give an upper bound on the number of colours required to extend a given vertex colouring of a graph to a total colouring. This shows that for any simple graph there is a total colouring using at most :d + 3 colours, where A is the maximum vertex degree.

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a nonlocally finite graph for w...

With any (not necessarily proper) edge k-colouring γ : E(G) −→ {1, . . . , k} of a graph G, one can associate a vertex colouring σγ given by σγ(v) = ∑ e∋v γ(e). A neighbour-sumdistinguishing edge k-colouring is an edge colouring whose associated vertex colouring is proper. The neighbour-sum-distinguishing index of a graph G is then the smallest k for which G admits a neighbour-sum-distinguishin...

this study attempts to investigate the origin of red pigments, time of reddening, and processes responsible for red colouring of the upper miocene age upper red formation, the most important clastic unit in central iran. mineralogical composition and textural properties of the studied sequences show that the upper red formation sequences were not red when deposited, although some parts of red p...

This study attempts to investigate the origin of red pigments, time of reddening, and processes responsible for red colouring of the Upper Miocene age Upper Red Formation, the most important clastic unit in central Iran. Mineralogical composition and textural properties of the studied sequences show that the Upper Red Formation sequences were not red when deposited, although some parts of red p...

We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a 2-out-colouring) is NP-complete. We show that for every choice of positive integers r, k there exists a k-strong bipartite tournament which needs at least r colours in...

In many applications of graph colouring the sizes of colour classes should not be too large. For example, in scheduling jobs (some of which could be performed at the same time), it is not good if the resulting schedule requires many jobs to occur at some specific time. An application of this type is discussed in [8]. A possible formalization of this restriction is the notion of equitable colour...

The following EREW PRAM algorithms for edge-colouring a general graph are presented: 1. an algorithm that finds a ðDþ dÞ-edge-colouring, 1pdoD; in Oððlog d þ ðD=dÞÞ log nÞ time, using n þ m processors; 2. an algorithm that finds a D1þe-edge-colouring, 0oeo1; in OðlogD log nÞ time, using nD1þe processors. r 2003 Elsevier Inc. All rights reserved.

There is a vast literature already in existence on colourings in graphs and designs. We refer the interested reader to [2, 7, 8, 11, 12]. A major application of such colourings is to sampling and scheduling problems. For an excellent consideration of designs for statistical purposes, see [13]; in [8], examples of graph colouring applications in scheduling are described. Let P be a point set and...