More Lower Bounds for Weak Sense of Direction: The Case of Regular Graphs

A graph G with n vertices and maximum degree G cannot be given weak sense of direction using less than G colours. It is known that n colours are always sufficient, and it was conjectured that just G + 1 are really needed, that is, one more colour is sufficient. Nonetheless, it has just been shown [2] that for sufficiently large n there are graphs requiring (n/ log n) more colours than G. In this paper, using recent results in asymptotic graph enumeration, we show not only that (somehow surprisingly) the same bound holds for regular graphs, but also that it can be improved to (n log log n/ log n). We also show that dG log log dG colours are necessary, where dG is the degree of G.

• Type: Conference
• Year: 2000
• Where: WDAG
• Authors: Paolo Boldi, Sebastiano Vigna
• Keywords: Algorithms, Asymptotic Graph Enumeration, Log DG Colours, N/ Log N, WDAG 2000