Holroyd, Fred and Watts, Ivor
(2010).
Overlap colourings and homomorphisms of graphs.
*The Quarterly Journal of Mathematics* [in press].

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## Abstract

An (*r*, λ) *overlap colouring* of a graph *G* has *r* colours at each vertex, any two adjacent vertices sharing exactly λ colours. A theory analogous to multichromatic and fractional chromatic theory is developed. In particular, all the overlap chromatic numbers of cycle graphs are computed. It is shown that if a graph *G* contains an odd cycle *C*_{2p+1} and has the same *p*-fold chromatic number as *C*_{2p+1}, then all its overlap chromatic numbers are the same as those of *C*_{2p+1}.
The *core* of a graph is the smallest induced subgraph to which it has a homomorphism. It is shown that some pairs of graphs with the same multichromatic numbers have different sets of overlap chromatic numbers, and that some graphs with non-isomorphic cores have the same sets of overlap chromatic numbers. (In particular, any non-bipartite series-parallel graph has the same overlap chromatic as its smallest odd cycle.) Thus classifying graphs by overlap chromatic properties is intermediate between classifying them by multichromatic properties and classifying them by cores.

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