# Overlap colourings and homomorphisms of graphs

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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 C2p+1 and has the same p-fold chromatic number as C2p+1, then all its overlap chromatic numbers are the same as those of C2p+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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