The Adaptable Chromatic Number and the Chromatic Number

We prove that the adaptable chromatic number of a graph is at least asymptotic to the square root of the chromatic number. This is best possible. Consider a graph G where each edge of G is assigned a colour from {1, ..., k} (this is not necessarily a proper edge colouring). A k-adapted colouring is an assignment of colours from {1, ..., k} to the vertices of G such that there is no edge with the same colour as both of its endpoints. In other words: in conventional graph colouring, each edge forbids its endpoints from both receiving the same colour, while in adaptable colouring, each edge is given one particular colour which it forbids its endpoints from both receiving. The adaptable chromatic number, χa(G), of a graph G is the minimum value of k such that every k-edge colouring of G can be completed into a k-adapted colouring. To be clear: an adapted colouring might not be proper. Eg. two adjacent vertices u, v can both receive the colour Red if the edge uv has a colour other than Red. It is not surprising that this natural variation on graph colouring has arisen in a variety of settings. Hell and Zhu[9] were the first to use the terminology adaptable colouring. But it was introduced independently as split colourings[4], emulsive colourings[2] and chromatic capacity[1]. Greene[7] was the first to conjecture the adaptable chromatic number grows with the chromatic number. He suggested that possibly χa(G) is always as high as Θ( √ χ(G)), noting that this would be best possible up to the multiplicative constant as Erdős and Gyárfás[4] had shown that (1 + o(1)) √ n ≤ χa(Kn) ≤ √ 2n. Huizenga[10] proved that the conjecture holds for almost all graphs. Zhou[14] proved the conjecture for every graph, showing that χa(G) ≥ Θ(log logχ(G)). Other work on adaptable colouring can be found in [3, 5, 6, 8, 11, 12, 13, 15]. Here, we give a very short proof of a bound that is tight up to second order terms: Theorem 1 χa(G) ≥ (1 + o(1)) √ χ(G). The tightness follows from the main theorem of [12] which showed that if G has maximum degree ∆ then χa(G) ≤ (1 + o(1)) √ ∆, thus improving the constant from the bound in [4] to ∗Dept of Computer Science, University of Toronto, [email protected]. Research supported by an NSERC Discovery Grant.