Backbone colourings of graphs

Abstract

a b s t r a c tConsider an undirected graph G and a subgraph of G, called H. A backbone k-colouring of (G, H) is a colouring f : V (G) → {1, 2, . . . , k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least 2. The minimum number k for which there is a backbone k-colouring of (G, H) is the backbone chromatic number BBC(G, H).We prove that every graph with chromatic number k has a proper k-colouring and a spanning tree T such that the colours on the endpoints of edges of T differ by exactly 1.As a corollary of this result, we show that for any graph G with χ (G) ≥ 4 there exists a spanning tree T of G such that BBC(G, T ) = χ (G).