Fractional chromatic number of distance graphs generated by two-interval sets

Let D be a set of positive integers. The distance graph generated by D , denoted by G ( Z , D ) , has the set Z of all integers as the vertex set, and two vertices x and y are adjacent whenever | x − y | ∈ D . For integers 1 < a ≤ b < m − 1 , define D a , b , m = { 1 , 2 , … , a − 1 } ∪ { b + 1 , b + 2 , … , m − 1 } . For the special case a = b , the chromatic number for the family of distance graphs G ( Z , D a , a , m ) was first studied by R.B. Eggleton, P. Erdős and D.K. Skilton [Colouring the real line, J. Combin. Theory (B) 39 (1985) 86–100] and was completely solved by G. Chang, D. Liu and X. Zhu [Distance graphs and T -coloring, J. Combin. Theory (B) 75 (1999) 159–169]. For the general case a ≤ b , the fractional chromatic number for G ( Z , D a , b , m ) was studied by P. Lam and W. Lin [Coloring distance graphs with intervals as distance sets, European J. Combin. 26 (2005) 25 1216–1229] and by J. Wu and W. Lin [Circular chromatic numbers and fractional chromatic numbers of distance graphs with distance sets missing an interval, Ars Combin. 70 (2004) 161–168], in which partial results for special values of a , b , m were obtained. In this article, we completely settle this problem for all possible values of a , b , m .