Edge-colouring of graphs and hereditary graph properties

Abstract

Edge-colourings of graphs have been studied for decades. We study edge-colourings with respect to hereditary graph properties. For a graph G, a hereditary graph property P and l ⩾ 1 we define $$X{'_{P,l}}$$ (G) to be the minimum number of colours needed to properly colour the edges of G, such that any subgraph of G induced by edges coloured by (at most) l colours is in P. We present a necessary and sufficient condition for the existence of $$X{'_{P,l}}$$ (G). We focus on edge-colourings of graphs with respect to the hereditary properties O k and S k , where O k contains all graphs whose components have order at most k+1, and S k contains all graphs of maximum degree at most k. We determine the value of $$X{'_{{S_k},l}}(G)$$ for any graph G, k ⩾ 1, l ⩾ 1, and we present a number of results on $$X{'_{{O_k},l}}(G)$$ .