General neighbour-distinguishing index via chromatic number

An edge colouring of a graph G without isolated edges is neighbour-distinguishing if any two adjacent vertices have distinct sets consisting of colours of their incident edges. The general neighbour-distinguishing index of G is the minimum number gndi ( G ) of colours in a neighbour-distinguishing edge colouring of G . Győri et al. [E. Győri, M. Horňák, C. Palmer, M. Woźniak, General neighbour-distinguishing index of a graph, Discrete Math. 308 (2008) 827–831] proved that gndi ( G ) ∈ { 2 , 3 } provided G is bipartite and gave a complete characterisation of bipartite graphs according to their general neighbour-distinguishing index. The aim of this paper is to prove that if χ ( G ) ≥ 3 , then ⌈ log 2 χ ( G ) ⌉ + 1 ≤ gndi ( G ) ≤ ⌊ log 2 χ ( G ) ⌋ + 2 . Therefore, if log 2 χ ( G ) ∉ Z , then gndi ( G ) = ⌈ log 2 χ ( G ) ⌉ + 1 .