Incidence coloring on hypercubes

Let ? i ( G ) and Δ ( G ) denote the incidence coloring number and the maximum degree of a graph G, respectively. An easy observation shows that ? i ( G ) ? Δ ( G ) + 1 . In this paper, we consider incidence coloring number on hypercubes Q n . Based on the technique of Hamming codes, we present algorithms to obtain upper bounds of ? i ( Q n ) for n in certain forms of integers. Let p , q be positive integers. We show that (1) ? i ( Q n ) = n + 1 if n = 2 p - 1 for p ? 1 ; (2) ? i ( Q n ) = n + 2 if the following conditions hold: (i) n = 2 p - 2 for p ? 2 ; (ii) n = 2 p + 2 q - 2 for p , q ? 1 ; (iii) n = 2 p + 2 q - 3 for p , q ? 2 ; and (3) ? i ( Q n ) ? n + 2 otherwise.