Abstract
Consider a graph G on kr vertices. We say that G is strongly r-colourable if for any partition of V (G) into parts V 1,..., V k, each of size r, G has a r-colouring such that every colour class contains exactly one vertex from each part (and so every part contains exactly one vertex of each colour). Equivalently, G is strongly r-colourable if for any graph G′ which is the union of k disjoint r-cliques on the same vertex set, χ(G U G) = r. A well-known conjecture of Erdős, recently proven by Fleischner and Steibitz [58], states that the union of a Hamilton cycle on 3n vertices and n vertex disjoint triangles on the same vertex set has chromatic number 3. In other words, C 3 n is strongly 3-colourable. Strongly r-colourable graphs are of interest partially because of their relationship to this problem, and also because they have other applications (see for example, Exercise 8.1).
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© 2002 Springer-Verlag Berlin Heidelberg
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Molloy, M., Reed, B. (2002). The Strong Chromatic Number. In: Graph Colouring and the Probabilistic Method. Algorithms and Combinatorics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04016-0_8
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DOI: https://doi.org/10.1007/978-3-642-04016-0_8
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