Circulant Distant Two Labeling and Circular Chromatic Number

Let G be a graph and d,d 0 be positive integers, d 0 d. An m-(d,d 0 )-circular distance two labeling is a function f : V (G) ! {0,1,2,···,m 1} such that |f(u) f(v)|m d if u and v are adjacent; and |f(u) f(v)|m d 0 if u and v are distance two apart, where |x|m := min{|x|,m| x|}. The minimum m such that there exists an m-(d,d 0 )-circular labeling for G is called the d,d0-number of G and denoted by d,d 0(G). The d,d 0-numbers for trees can be obtained by a first-fit algorithm. In this article, we completely determine the d,1-numbers for cycles. In addition, we show connections between generalized circular distance labeling and circular chromatic number.