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Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring...
For properties of graphs P1 and P2 a vertex (P1,P2)-partition of a graph G is a partition (V1,V2) of V(G) such that each subgraph G[Vi] induced by Vi has property Pi,i=1,2. The class of all vertex (P1,P2)-partitionable graphs is denoted by P1∘P2. An additive hereditary property R is reducible if there exist additive hereditary properties P1 and P2 such that R=P1∘P2, otherwise it is irreducible. For...
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